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Supersingular elliptic curves and twisting endomorphisms

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Sarah Arpin, Josep M. Miret, Jordi Pujolàs, Javier Valera

Number Theory · Submitted 30 Jun 2026 · 29 references

Pith's verdict: The generalization to O-oriented curves holds up with the needed algebraic checks, and the search algorithm carries over directly.

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arXiv ID2606.31687 (version 1)
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CitationJavier Valera, Jordi Pujol\`as, Josep M. Miret, Sarah Arpin. "Supersingular elliptic curves and twisting endomorphisms." arXiv:2606.31687 (2026). https://pith.science/paper/CH22KX3U
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The paper

The paper claims

An algorithm locates supersingular elliptic curves over F_p that admit twisting endomorphisms of prime degree and computes bases for their full endomorphism rings.

The reading

The paper extends the definition of twisting endomorphisms to supersingular elliptic curves that carry an orientation by an order. It supplies a search procedure that identifies curves over the finite field with a prime-degree twisting endomorphism. The same data is then used to produce a basis for the complete endomorphism ring. A sympathetic reader would care because the endomorphism ring governs the arithmetic behavior and structural invariants of these curves.

What carries the argument

The twisting endomorphism of prime degree ℓ in the O-oriented setting, which both locates the curve and supplies generators for its endomorphism ring basis.

Extended reading notes

Core claim

We generalize the notion of twisting endomorphisms to the setting of O-oriented supersingular elliptic curves. We give an algorithm to find supersingular elliptic curves over F_p with a twisting endomorphism of prime degree ℓ, and we use it to compute a basis of their full endomorphism rings.

Load-bearing premise

The algebraic properties that let twisting endomorphisms serve as search targets and ring generators survive the move from the unoriented case to the oriented case.

Editorial extensions

If this is right

  • The search procedure terminates with at least one curve possessing the stated twisting endomorphism.
  • The twisting endomorphism together with the Frobenius endomorphism generate the entire endomorphism ring.
  • The method applies uniformly for any order O and any prime ℓ compatible with the orientation.
  • Explicit bases for the rings become computable once the curves are located.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the procedure runs in reasonable time, it offers a direct route to explicit examples of curves whose endomorphism rings are known in advance.
  • The same search idea could be tested on endomorphisms of composite degree or on orientations by different orders.
Original abstract

We generalize the notion of twisting endomorphisms, first defined by Castryck-Panny-Vercauteran, to the setting of $\mathcal{O}$-oriented supersingular elliptic curves. We give an algorithm to find supersingular elliptic curves over $\mathbb{F}_p$ with a twisting endomorphism of prime degree $\ell$, and we use it to compute a basis of their full endomorphism rings.

Figures
Figure 1. O-twisting endomorphism diagram to accompany Example 2.10. At the top is the factorisation of the O-twisting endomorphism and the corresponding image orienta￾tions. Below, the image ι(r) of the generator of O is given, as an element of the quaternion order M ∼= End(E). suppose η corresponds to i+j and η ′ corresponds to −i−j under the identi￾fication in (5). Let K := Q(r)/(r 2 + 31) be the number field of discriminant −31, and let O := Z[r] denote the order in K generated by r. The two elements η, η′ are the two choices for O-orientations on E, say: ι : K ,→ End(E) ⊗Z Q ι(r) = η. I
Figure 2. Supersingular isogeny graph over F439 with degree 2,5, and 11 isogenies depicted as edges. We let t = tr(α ◦ β) and we impose maximality by solving tr([1] ◦ [1]) tr([1] ◦ α) tr([1] ◦ β) tr([1] ◦ α ◦ β) tr(α ◦ [1]) tr(α ◦ α) tr(α ◦ β) tr(α ◦ α ◦ β) tr(β ◦ [1]) tr(β ◦ α) tr(β ◦ β) tr(β ◦ α ◦ β) tr(α ◦ β ◦ [1]) tr(α ◦ β ◦ α) tr(α ◦ β ◦ β) tr(α ◦ β ◦ α ◦ β) = 2 0 0 t 0 −10 t 0 0 t −2m 0 t 0 0 t 2 − 10m = −(t 2 − 20m) 2 = −4392 29

Figures reproduced from arXiv:2606.31687 by the authors.

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