For every n ≥ 2, the number of n-dimensional rational tori with Artin conductor at most X is bounded by X^{exp(C(log n)^2)}, for an absolute constant C.
Borel, Introduction to arithmetic groups, Univ
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An upper bound for counting algebraic tori over $\mathbb{Q}$ by Artin conductor
For every n ≥ 2, the number of n-dimensional rational tori with Artin conductor at most X is bounded by X^{exp(C(log n)^2)}, for an absolute constant C.