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curve #6232

y2 + xy = x3 − x2 − 26239874541x + 1636045046698553
a-invariants
[1, -1, 0, -26239874541, 1636045046698553]
c-invariants
[1259513977977, -1413537252534648909]
rank (lower bound)
≥ 7
torsion subgroup
ℤ/3ℤ
conductor (N)
1392454680029442
discriminant (Δ)
-14755328331716680838495722116 ☆ record for ℤ/3ℤ torsion, rank ≥ 7
Faltings height
4.5816
naive height
83.5853
primes of bad reduction
2, 3, 37, 61, 89, 103, 233, 1783
regulator
14165.35669738120207828170506903632271019595591251202
submitted by
Lewis Combes
submitted at
last updated

Witness: 7 independent points log in to add more points →

Commentary

E has torsion Z/3Z and rank exactly 7. It is isomorphic to the Tate form y^2 + 1047xy - 2301807y = x^3 (a1 = 1047, a3 = -2301807), in which (0,0) is a point of order 3. Every curve with a rational point of order 3 has such a form y^2 + a1 xy + a3 y = x^3, and (a1, a3) and (a1/u, a3/u^3) give isomorphic curves (x -> u^2 x, y -> u^3 y), so the curve is determined by t = a3/a1^3: it is the member t = -767269/382576941 (in lowest terms) of the one-parameter Z/3 family y^2 + xy + ty = x^3. The submitted a-invariants are the global minimal model of this curve (the Tate form above is already minimal; the two differ by a translation), on which the point (0,0) of the Tate form is (91351, -1196579). Rank exactly 7: the seven points listed are independent (regulator 14165.357), and the 2-Selmer group has rank 7 (E has no rational 2-torsion; root number -1), so the rank is at most 7. The Selmer computation assumes GRH for the class groups of the cubic fields involved. Confirmed in PARI/GP (ellrank [7, 7]). It beats the previous T[3] rank-7 records in discriminant (#1661, https://elliptic-rank.icarm.cloud/curve/1661). Search scoped and carried out by Claude code Opus 5.5

last edited by Lewis Combes at · history

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