{"id":"53c846dd-2bd2-4a46-89c9-7d0f2b0de67a","arxiv_id":"2505.20479","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For elliptic curves over Q with rational isogenies of various degrees, the Tamagawa number is shown to have only prescribed prime divisors, and infinite families with small prescribed Tamagawa numbers are constructed.","lead":"This paper classifies which primes can divide the Tamagawa numbers of elliptic curves over the rationals that carry rational isogenies or torsion, and constructs infinite families with very small Tamagawa numbers. Since Tamagawa numbers appear in the Birch and Swinnerton-Dyer conjecture, the results sharpen a central invariant of elliptic curves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The most load-bearing gap is the unverified LMFDB reduction-type data for the base curves in Propositions 2.10 and 2.12, which directly support Theorem 1.1(iv)-(v); a single misquoted Kodaira type at p=2 or 3 could break the claimed prime restrictions.","rationale":"The central theorem asserts very specific prime restrictions on Tamagawa numbers for curves with isogenies of degree N=14,15,21,27. For these composite N, the proof classifies possible j-invariants and reduces to quadratic twists of a small set of base curves. The admissible Tamagawa numbers at the small primes 2,3,5,7 are then computed from the Kodaira types of those base curves. These Kodaira types are quoted from LMFDB with no computation or independent reference in Propositions 2.10 and 2.12 (and likewise in 2.8 and 2.14). This is the single most load-bearing assumption: if any quoted type is wrong, the conclusion for that N changes. A concrete Tate-algorithm check on the listed curves would settle the issue. The reader's weakest-assumption field points to exactly this, and I agree. I also note the false claim v_q(j)>0 in the proof of Proposition 2.16; since the required inequality is v_q(j)≥0 and the bound c_q≤4 follows either way, this is a correctable typo rather than a fatal flaw. No other concern outweighs the local-data gap, so the CONDITIONAL verdict stands without change.","tokens_in":16320,"tokens_out":15478,"duration_ms":146036,"concrete_test":"Run Tate's algorithm (e.g., in SageMath) on the minimal Weierstrass equations of the curves 50.a1-50.a4, 162.c1-162.c4, 49.a1-49.a2, and 27.a1 obtained from LMFDB, computing the Kodaira symbols and Tamagawa numbers at p=2 and p=5 for the 50.a curves, p=2 and p=3 for the 162.c curves, p=7 for the 49.a curves, and p=3 for 27.a1. Compare with the reduction types asserted in Propositions 2.8, 2.10, 2.12, and 2.14. If every asserted type matches, the concern is resolved; if any differs, recompute the twist table and check whether Theorem 1.1(iv)/(v) still holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1(iv) and (v) rest on Propositions 2.10 and 2.12, which assert the local reduction types of the base curves 50.a1-50.a4 and 162.c1-162.c4 at p=2,3,5. These assertions are quoted directly from LMFDB without any Tate-algorithm computation or citable reference beyond the database. Since every curve with the relevant j-invariant is a quadratic twist of one of these base curves, the possible Tamagawa numbers at p=2,3,5 are derived by applying Table 2 to those base types. If, for example, 50.a4 had a Kodaira type at p=2 different from I15—say a wild type giving c2=17 or In with n having a prime factor outside {2,3,5}—then Proposition 2.10(iii) and Theorem 1.1(v) could fail. Similarly, if 162.c1-c4 had different types at p=2 or p=3, the 7-part in Theorem 1.1(iv) could be wrong. The database is likely correct, but the proof as written gives no independent verification. Separately, the proof of Proposition 2.16 contains the false assertion that v_q(j)>0 for every q≠p; the needed inequality is v_q(j)≥0, and the subsequent bound c_q≤4 follows either way, so this is a typo rather than a load-bearing flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Tamagawa numbers c(E) of elliptic curves over Q that have rational cyclic isogenies or torsion points. Theorem 1.1 claims restrictions on the prime divisors of c(E) according to the degree N of a cyclic Q-rational isogeny, with N ranging over the Mazur-Kenku list. Section 2 proves these restrictions using reduction-type classifications from the author's earlier work [23] and quadratic-twist tables, and also produces infinite families with isogenies of degree 5, 7, and 13 whose Tamagawa numbers are of the form 2^n 3^m. Section 3 constructs infinite families with a rational point of order 4 or 5 and small Tamagawa numbers, and proves that for N'=4,5,6,7,8,9,10,12 any prime can divide c(E) for infinitely many curves with an N'-isogeny. Section 4 gives bounds on Tamagawa numbers of specializations of elliptic surfaces in terms of the discriminant of a minimal Weierstrass equation, including an abc-conditional result producing infinitely many specializations with Tamagawa number 1.","tokens_in":16576,"tokens_out":25115,"duration_ms":251591,"significance":"If the issues below are corrected, the paper makes a useful contribution to a natural question: which primes divide Tamagawa numbers of elliptic curves with prescribed rational isogenies, and how small can these Tamagawa numbers be. The paper uses standard tools (Tate's algorithm, quadratic-twist tables, the Mazur-Kenku classification, sieve results) in a transparent way, and several of the construction results in Sections 3 and 4 are concrete and unconditional. The main theorem's prime-divisor restrictions, if valid, would refine earlier work of Lorenzini and Trbović. The dependence on LMFDB local data is a legitimate gap in exposition, but it is likely fillable by direct Tate-algorithm computations. The paper is not circular: it relies on earlier published work [23] and external classifications.","major_comments":[{"comment":"The exponent condition n≥1 in Theorem 1.1(iv)-(v) is not justified and appears to be contradicted by the local data in the same paper. In Proposition 2.10, the base curve E2 = 50.a1 has reduction type I3 modulo 2 and IV modulo 5, and the proposition states that all curves in that isogeny class have conductor 2·5^2. Since the only bad primes are 2 and 5, the local Tamagawa numbers are c2∈{1,3} (odd, by Table 1 for I3) and c5∈{1,3} (odd, by Table 1 for IV). Thus the global c(E2) is odd, contradicting n≥1 in Theorem 1.1(v). Similarly, the curve 162.c2 named in Remark 2.13 has c2=21 and reduction type II at 3 (so c3=1), with conductor 2·3^4, giving c(E)=21, which has no factor 2 and contradicts n≥1 in Theorem 1.1(iv). Please correct the exponent range, most likely to n≥0, or provide a separate argument showing that these curves do not have the required global Tamagawa numbers.","section":""},{"comment":"The reduction types of the base curves 50.a1-50.a4 at p=2,5 and 162.c1-162.c4 at p=2,3 are quoted from LMFDB [17] without an independent computation or a citable proof. These local data are load-bearing for Theorem 1.1(iv)-(v), because the twist argument transfers only these base types to all twists. Please include either a Tate-algorithm computation for each base curve, or explicit minimal Weierstrass equations together with the resulting Kodaira types, so that the proof is self-contained and the reader can verify the claimed values c2=1,3,5,15 for N=15 and c2=1,2,3,4,7,21 for N=21.","section":"Propositions 2.10 and 2.12"},{"comment":"The proof contains the assertion \"v_q(j(E_{t0,ℓ})) > 0 for every prime q ≠ p\" after taking t0=p. This is false: for the rational function F_ℓ(t), with denominator t, one has v_q(j) ≥ 0 for all q ≠ p, with equality for most q. The desired bound c_q(E) ≤ 4 follows from [28, Corollary 9.2] already under v_q(j) ≥ 0, so the error is local and does not destroy the conclusion, but the inequality must be corrected to v_q(j) ≥ 0.","section":"Proposition 2.16, proof"}],"minor_comments":[{"comment":"The first sentence of the proof refers to an isogeny of degree 14, but the proposition concerns degree 27; this is a typo.","section":"Proposition 2.14, proof"},{"comment":"The concluding claim \"c(E)=2^n3^m, for some n≥2\" does not follow from the listed possibilities (i)-(iv), since c2=1, c5=2, and all other local factors equal to 1 would give n=1. Please verify whether the intended bound is n≥1 or whether an additional source of a factor 2 is forced by [23, Theorem 3.7].","section":"Proposition 2.4, conclusion"},{"comment":"The wording \"If p ̸= 2, ℓ is a prime\" should read \"If p is a prime different from 2 and ℓ\".","section":"Proposition 2.1(i)"},{"comment":"The inequality \"c_p(E_n) ≤ v_p(E_n)\" should refer to v_p(Δ(n)), not v_p(E_n).","section":"Theorem 4.3, proof"},{"comment":"The sentence \"Let q be a divisor of q\" is a typo; it should refer to a prime divisor q of d.","section":"Remark 2.15"},{"comment":"There are several grammatical slips (e.g., \"we will show a slightly statement\" in the proof of Proposition 3.5) and minor notation inconsistencies in Section 2; these should be cleaned up during revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the unjustified n≥1 bound in Theorem 1.1(iv)-(v), which appears to be contradicted by the paper's own base-curve examples. This is likely a simple exponent correction rather than a fatal flaw, but it must be addressed because Theorem 1.1 is the central result. The LMFDB verification request is standard for a number theory paper and should be straightforward to satisfy. The other mathematical concerns are local and fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is a genuine step forward on Tamagawa numbers of rational elliptic curves with isogenies. The main result, Theorem 1.1, gives a nearly complete description of the primes dividing the global Tamagawa number for every cyclic rational isogeny degree except the small ones (4,5,6,7,8,9,10,12), where the author shows the question is unbounded. That classification is new, and the paper also contains unconditional infinite families with prescribed small Tamagawa numbers (Theorem 1.2) and a general bound on specializations of elliptic surfaces (Theorem 1.3). The methods are standard—Tate's algorithm, modular curve parametrizations, quadratic twist tables, and sieve results—but the results are not in the prior literature.\n\nThe writing is mostly careful. Conditional results (Proposition 2.18, Theorem 4.3) are clearly marked as depending on finiteness of Tate-Shafarevich or abc, and the author’s earlier theorems [23] are used as black boxes that are proved independently. No circularity.\n\nThe soft spots are real but fixable. The reduction types for curves 50.a1–50.a4 and 162.c1–162.c4 at p=2,3,5 are quoted from LMFDB without independent computation. These are load-bearing for Propositions 2.10 and 2.12, and hence for Theorem 1.1(iv) and (v). The database is very likely right, but the proof as written has a gap. A referee should ask for the Tate algorithm output or a citable reference. Also, the proof of Proposition 2.16 says v_q(j)>0 for every q≠p, which is false; the correct statement is v_q(j)≥0. The conclusion c_q(E)≤4 holds either way, so it is a typo rather than a flaw. There are a few other typos (e.g., \"slightly statement\" in Proposition 3.5, \"Let q be a divisor of q\" in Remark 2.15) that a light edit will catch.\n\nOverall, I believe the main theorems hold, and the paper deserves a serious referee. It is aimed at number theorists interested in Tamagawa numbers, isogenies, and BSD. I would not desk-reject it. Send it to review, ask for verification of the LMFDB data and cleanup of the typos.\n\nBest,","headline":"Genuinely new classification of Tamagawa primes for rational isogenies, with two fixable gaps; deserves referee time.","tokens_in":17170,"tokens_out":3345,"would_cite":true,"duration_ms":31214,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11G07","11G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Cyclic rational isogenies of specified degrees restrict the global Tamagawa number of an elliptic curve over $\\mathbb{Q}$ to a very small set of prime divisors, and infinite families attain the small values.","keywords":["Tamagawa numbers","elliptic curves over Q","rational isogenies","torsion points","reduction types","quadratic twists","elliptic surfaces","BSD formula"],"falsifier":"Recompute, with the standard local reduction algorithm, the reduction types at $p=2$ and $p=3$ of the four degree-$21$ base curves of conductor $162$ and the four degree-$15$ base curves of conductor $50$; if the degree-$21$ types are not $(I_3,I_1,I_{21},I_7)$ at $p=2$ and $(II,II^*,II,II^*)$ at $p=3$, or the degree-$15$ types are not $(I_1,I_3,I_5,I_{15})$ at $p=2$, the corresponding part of Theorem 1.1 fails. A single curve with a rational $21$-isogeny and $c_3(E)>3$ or $c_2(E)>21$ would also disprove part (iv).","tokens_in":16057,"feed_emoji":"🧮","tokens_out":21614,"duration_ms":214460,"temperature":0.7,"pith_summary":"Tamagawa numbers measure, prime by prime, how an elliptic curve degenerates at bad primes; the global Tamagawa number $c(E)$ is their product. This paper proves that if $E/\\mathbb{Q}$ has a cyclic $\\mathbb{Q}$-rational isogeny of degree $N$ in a list of possible degrees, then $c(E)$'s prime divisors are severely restricted: for $N=14,19,43,67,163$ one gets $c(E)=2^n$; for $N=11,27,37$ one gets $2^n3^m$ with at most one factor $3$; for $N=17,21,15$ one extra prime appears at most to the first power; and for $N=9$ the power of $3$ is capped by $3^3$. The paper also shows that for rational torsion points of order $4,5,6,7,8,9,10,12$ no such restriction is possible, since every prime can be made to divide $c(E)$ for infinitely many curves. On the positive side it constructs infinite families with small Tamagawa numbers: order-$4$ torsion with $c(E)\\in\\{4,8,12\\}$, order-$5$ torsion with $c(E)\\le 30$, and isogenies of degree $5,7,13$ with $c(E)=2^n3^m$. These results matter because $c(E)$ is one of the factors in the BSD formula, so controlling it controls a global arithmetic invariant for whole families of curves.","feed_headline":"Isogenies pin Tamagawa numbers to tiny prime sets","feed_subtitle":"The restrictions limit the BSD quotient and leave only 2s and a few 3s, 5s, 7s, or 17s in the Tamagawa number.","key_machinery":"The carrying mechanism is the reduction-type dictionary for elliptic curves with $N$-isogenies. The modular curve $X_0(N)$ parametrizes pairs $(E,C)$ with a cyclic rational $N$-isogeny; for each $N$ in Theorem 1.1 its rational points give a finite list of $j$-invariants, and none are $0$ or $1728$, so every such curve is a quadratic twist of one of finitely many base curves. Two tables do the work: the table from the standard local reduction algorithm converting reduction types into local Tamagawa numbers (for example split $I_n$ gives $n$, non-split $I_n$ gives $1$ or $2$, $III$ gives $2$, additive types give $1$ or $3$), and the quadratic-twist table saying how reduction types swap at primes dividing the twist. For the specialization results, the mechanism is different: sieve theorems guarantee that the discriminant polynomial $\\Delta(T)$ of a one-parameter family $E/\\mathbb{Q}(T)$ takes values with few prime factors, so that specializations $E_n$ have only a few primes of bad reduction and $c(E_n)$ is bounded by an explicit expression in $\\deg\\Delta$.","core_discovery":"The central claim of the paper is Theorem 1.1. Let $E/\\mathbb{Q}$ be an elliptic curve with a cyclic $\\mathbb{Q}$-rational isogeny of degree $N$. Then: for $N=14,19,43,67,163$, $c(E)=2^n$ with $n\\ge1$; for $N=11,27,37$, $c(E)=2^n3^m$ with $m\\in\\{0,1\\}$; for $N=17$, $c(E)=2^n3^m17^k$ with $m,k\\in\\{0,1\\}$ and $n\\ge1$; for $N=21$, $c(E)=2^n3^m7^k$ with $m\\in\\{0,1,2\\}$, $k\\in\\{0,1\\}$, $n\\ge1$; and for $N=9$ or $15$, $c(E)=2^n3^m\\ell^k$ with $m\\in\\{0,1,2\\}$, $k\\in\\{0,1\\}$, where $\\ell=3$ or $5$ respectively. The proof shows that for each of these degrees every local Tamagawa factor $c_p(E)$ can only be $1,2,4$, with a single possible factor $3$, $5$, $7$, or $17$ in specified cases. It also proves that the remaining torsion orders $4,5,6,7,8,9,10,12$ admit no such prime bound, and supplies infinite families where $c(E)$ is as small as the constraints allow.","pith_inferences":["The same finite-$j$-invariant plus quadratic-twist strategy could be applied to other isogeny degrees with a known table of $j$-invariants, such as degree $25$, to predict analogous prime restrictions; the paper does not attempt this.","The sieve bound in Theorem 1.3 depends on an explicit but non-optimal constant $s$ from the almost-prime theorem; a sharper sieve would shrink the bound and could make the difference $C_2-C_1$ in Question A depend only on $\\deg\\Delta$.","For most of the small torsion orders, Proposition 3.5 shows any prime can be forced into $c(E)$, which indicates that Theorem 1.1's restrictive list is close to optimal.","The parity formula $v_7(c(E))\\equiv v_7(c(E'))+\\operatorname{rk}E/\\mathbb{Q}+1\\pmod2$, conditional on the standard finiteness assumption for the relevant Galois cohomology group, offers a local-data route to rank-parity statements for curves with $7$-torsion and semistable reduction at $7$."],"forward_implications":["For every curve with a rational isogeny of degree $19,43,67$, or $163$, the global Tamagawa number is a power of $2$, so the BSD quotient $c(E)/|E(\\mathbb{Q})_{\\mathrm{tors}}|$ has no odd primes in its Tamagawa part.","For isogeny degrees $11,27,37$ the $3$-part of $c(E)$ is at most $3$; for degrees $17,21,15$ the extra prime appears at most to the first power and the $3$-part at most $9$; for degree $9$ the $3$-part is at most $27$.","A rational $14$-isogeny forces $c(E)$ to be even, refining the earlier result that $2\\mid c(E)$ for such curves.","Infinitely many $j$-distinct curves with isogeny degree $5,7$, or $13$ have $c(E)=2^n3^m$, and infinitely many curves with a rational point of order $4$ (respectively $5$) have $c(E)\\in\\{4,8,12\\}$ (respectively $c(E)\\le30$).","For a non-isotrivial one-parameter family of elliptic curves whose discriminant has few roots modulo every prime, infinitely many specializations satisfy the explicit bound $c(E_n)\\le16(\\log_2(m)+s\\deg\\Delta)^{d(m)+s}$, and under the $abc$-conjecture with squarefree discriminant values there are infinitely many specializations with $c(E_n)=1$."],"supporting_citations":[{"why":"Classifies the possible prime degrees of rational isogenies, fixing the list $N=11,\\ldots,163$.","marker":"[21]"},{"why":"Completes the classification of rational isogeny degrees to composite degrees such as $21,25,27,37,43,67$, and $163$.","marker":"[15]"},{"why":"Supplies the possible reduction types at primes for curves with $N$-isogenies for $N=11,17,19,37,43,67$, and $163$.","marker":"[23]"},{"why":"Provides the table of $j$-invariants of $\\mathbb{Q}$-rational points on $X_0(N)$ used to identify the base twist classes for $N=14,15,21$, and $27$.","marker":"[20]"},{"why":"Gives the table of how reduction types change under quadratic twists, which carries the twisting step.","marker":"[5]"},{"why":"Supplies the local reduction algorithm table linking reduction types to local Tamagawa numbers, plus the relevant Néron model facts.","marker":"[28]"},{"why":"Provides the definition of the Tamagawa number, the BSD formula, and the torsion facts used throughout.","marker":"[29]"},{"why":"Quoted as the database of curve invariants giving the reduction types of the base curves at the small primes $2,3$, and $5$.","marker":"[17]"},{"why":"Supplies the sieve theorem on almost-prime values of polynomials used to bound Tamagawa numbers of specializations.","marker":"[12]"},{"why":"Provides the explicit one-parameter families of curves with rational points of order $4$ and $5$ used in Theorem 1.2.","marker":"[13]"}],"fun_headline_variants":["Isogenies restrict Tamagawa numbers to powers of 2 and rare small primes","Tamagawa numbers of isogenous elliptic curves: mostly 2^n, sometimes 3,5,7,17","Elliptic curves with isogenies have Tamagawa numbers with primes only 2,3,5,7,17","For many isogeny degrees, Tamagawa numbers are 2^n times 1,3,5,7, or 17","Isogenies force Tamagawa primes into {2,3,5,7,17} for many degrees"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the quoted reduction types of the small base curves at primes $2$, $3$, and $5$—how each curve degenerates modulo those primes—are correct as taken from the database; the paper does not recompute them, and a single error there would break the corresponding restriction in Theorem 1.1.","fun_headline_variants_meta":{"raw":{"variants":["Isogenies restrict Tamagawa numbers to powers of 2 and rare small primes","Tamagawa numbers of isogenous elliptic curves: mostly 2^n, sometimes 3,5,7,17","Elliptic curves with isogenies have Tamagawa numbers with primes only 2,3,5,7,17","For many isogeny degrees, Tamagawa numbers are 2^n times 1,3,5,7, or 17","Isogenies force Tamagawa primes into {2,3,5,7,17} for many degrees"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002002,"raw_usage":{"total_tokens":7814,"prompt_tokens":955,"completion_tokens":6859,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":6721}},"tokens_in":571,"tokens_out":6859,"duration_ms":53969,"temperature":1.0,"reasoning_tokens":6721,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:53:57.132311+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute, with the standard local reduction algorithm, the reduction types at $p=2$ and $p=3$ of the four degree-$21$ base curves of conductor $162$ and the four degree-$15$ base curves of conductor $50$; if the degree-$21$ types are not $(I_3,I_1,I_{21},I_7)$ at $p=2$ and $(II,II^*,II,II^*)$ at $p=3$, or the degree-$15$ types are not $(I_1,I_3,I_5,I_{15})$ at $p=2$, the corresponding part of Theorem 1.1 fails. A single curve with a rational $21$-isogeny and $c_3(E)>3$ or $c_2(E)>21$ would also disprove part (iv).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies the possible prime degrees of rational isogenies, fixing the list $N=11,\\ldots,163$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Completes the classification of rational isogeny degrees to composite degrees such as $21,25,27,37,43,67$, and $163$."},{"cited_title":"Melistas","cited_arxiv_id":null,"evidence_quote":"Supplies the possible reduction types at primes for curves with $N$-isogenies for $N=11,17,19,37,43,67$, and $163$."},{"cited_title":"Lozano-Robledo","cited_arxiv_id":null,"evidence_quote":"Provides the table of $j$-invariants of $\\mathbb{Q}$-rational points on $X_0(N)$ used to identify the base twist classes for $N=14,15,21$, and $27$."},{"cited_title":"Comalada","cited_arxiv_id":null,"evidence_quote":"Gives the table of how reduction types change under quadratic twists, which carries the twisting step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the local reduction algorithm table linking reduction types to local Tamagawa numbers, plus the relevant Néron model facts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the definition of the Tamagawa number, the BSD formula, and the torsion facts used throughout."},{"cited_title":"The L-functions and modular forms database.https://www.lmfdb.org, 2025","cited_arxiv_id":null,"evidence_quote":"Quoted as the database of curve invariants giving the reduction types of the base curves at the small primes $2,3$, and $5$."},{"cited_title":"Halberstam and H.-E","cited_arxiv_id":null,"evidence_quote":"Supplies the sieve theorem on almost-prime values of polynomials used to bound Tamagawa numbers of specializations."},{"cited_title":"Husemöller","cited_arxiv_id":null,"evidence_quote":"Provides the explicit one-parameter families of curves with rational points of order $4$ and $5$ used in Theorem 1.2."}],"review_version":1}