A data structure representing isomoprhism classes of elliptic curves over ℚ. Every elliptic curve over ℚ is isomorphic to one with Weierstrass equation y² = x³ + Ax + B, and the pair (A,B) is unique if it satisfy the reduced condition below. See Section 5.1 in [PPVW2016].

The rank of an elliptic curve over a number field is always finite by the Mordell–Weil theorem. Consequently, the rank is always finite, so finrank ℤ E⟮K⟯ = 0 really means that the group of rational points is torsion, not that it is of infinite rank.

Convert the structure RatEllipticCurve to a Weierstrass curve.

The rank of an elliptic curve over ℚ.

The naïve height of an elliptic curve over ℚ.

The set of elliptic curves over ℚ with naïve height less than or equal to a given height.

Formula (5.1.1) of [PPVW2016]: The number of elliptic curves over ℚ with naïve height at most H is asymptotically 2^(4/3)*3^(-3/2)/ζ(10) * H^(5/6).

Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their heights, then 50% of the curves have rank 0 and 50% have rank 1. See p. 28 of https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Bhargava.pdf.

Theorem 3 of [BS2013]: when elliptic curves over ℚ are ordered by height, their average rank is < .885.

Theorem 4 of [BS2013]: when elliptic curves over ℚ are ordered by height, a density of at least 83.75% have rank 0 or 1.

Theorem 5 of [BS2013]: when elliptic curves over ℚ are ordered by height, a density of at least 20.62% have rank 0.

The quadratic twist $E^d$ of $E : y^2 = x^3 + Ax + B$ by an integer $d$, given by the Weierstrass equation $y^2 = x^3 + d^2Ax + d^3B$. For $d \neq 0$ this is the twist of $E$ corresponding to the quadratic field $\mathbb{Q}(\sqrt{d})$. See Section 1 of [Smith2025].

The twist of $E$ by $d = 1$ is $E$ itself.

The quadratic twist of an elliptic curve over $\mathbb{Q}$ by a nonzero $d$ is again an elliptic curve.

The naïve height of the Weierstrass model $E^d$ is $|d|^6$ times the naïve height of $E$. Ordering the twists of $E$ by $|d|$ is therefore the same as ordering them by naïve height.

The rank of the quadratic twist $E^d$ of an elliptic curve $E$ over $\mathbb{Q}$.

The set of nonzero integers $d$ with $|d| \leq H$, which index the quadratic twists $E^d$ ordered by $|d|$. By naiveHeight_quadraticTwist this is the same as ordering them by height.

There are $2H$ nonzero integers $d$ with $|d| \leq H$. This is the normalisation used in [Smith2025].

Goldfeld's conjecture ([Goldfeld1979], Conjecture B), in the form stated in the introduction of [Smith2025]: when the quadratic twists $E^d$ of an elliptic curve $E$ over $\mathbb{Q}$ are ordered by $|d|$, 50% of them have rank $0$ and 50% have rank $1$. See goldfeld_conjecture.variants.two_le for the remaining case $2 \leq r$.

Goldfeld states the conjecture for the analytic rank of $E^d$, which the Birch and Swinnerton-Dyer conjecture predicts is equal to the Mordell–Weil rank used here. Corollary 1.2 of [Smith2025] proves that the Birch and Swinnerton-Dyer conjecture for the quadratic twist family of $E$ implies Goldfeld's conjecture for $E$.

Goldfeld's conjecture ([Goldfeld1979], Conjecture B), in the form stated in the introduction of [Smith2025], in the case $2 \leq r$: a density $0$ of the quadratic twists $E^d$ of an elliptic curve $E$ over $\mathbb{Q}$ have rank $r$.

Goldfeld's conjecture with $d$ restricted to squarefree integers, which is how it is usually stated informally: 50% of the quadratic twists of $E$ have rank $0$ and 50% have rank $1$. Every quadratic twist of $E$ is isomorphic to $E^d$ for a unique squarefree $d$, so the squarefree $d$ enumerate the distinct twists without repetition.

From [PPVW2016], Section 3.1: "from the mid-1960s to the present, it seems that most experts conjectured unboundedness."

From [PPVW2016], Section 8.2: "Our heuristic predicts (a) All but finitely many E ∈ ℰ satisfy rk E(ℚ) ≤ 21". In other words, there are only finitely many elliptic curves over ℚ (up to isomorphism) with rank greater than 21. Notice that this contradicts the previous conjecture.

[PPVW2016] 8.2(b): for 1 ≤ r ≤ 20, the number of elliptic curves over ℚ with rank r and naïve height at most H is asymptotically H ^ ((21 - r) / 24 + o(1)). Note: ℰ_H in 8.2(b) should be ℰ_{≤H}, see the statement of Theorem 7.3.3. When r = 1, the exponent is 20 / 24 = 5 / 6, which agrees with the exponent in card_heightLE_div_pow_five_div_six_tensto and is consistent with half_rank_zero_and_half_rank_one.

[PPVW2016] 8.2(c): the number of elliptic curves over ℚ with rank ≥ 21 and naïve height at most H is asymptotically at most H ^ o(1).