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curve #6456

y2 = x3 + x2 − 1678544765195830521582823433x + 18171698656686596939062151894922391606263
a-invariants
[0, 1, 0, -1678544765195830521582823433, 18171698656686596939062151894922391606263]
c-invariants
[80570148729399865035975524800, -15700347639377703176242075636403162200960000]
rank (lower bound)
≥ 11
torsion subgroup
ℤ/2ℤ × ℤ/2ℤ
conductor (N)
16492761487975778066271897405280161600
discriminant (Δ)
160025544932199927450311330216051547754170543357206105737180807237887269273664000000
Faltings height
14.6702
naive height
199.6768
primes of bad reduction
2, 3, 5, 7, 11, 13, 29, 43, 47, 61, 67, 83, 109, 113, 127, 149, 181, 223, 241, 271, 281
regulator
195681649.910594470722839983605954940555232356141818614176082
submitted by
The Wren
submitted at
last updated

Witness: 11 independent points log in to add more points →

Commentary

Z/2xZ/2 X400a family y^2=x(x+A)(x+B), with A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q), at T=p/q=-473/669. Family equations as posted by wgxli with curve #4252: https://elliptic-rank.icarm.cloud/curve/4252 Section formulas from the Layr-Labs elliptic-rank-challenge-public repository, submission/research/x400a-family.gp, commit 7482bf9: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/7482bf9/submission/research/x400a-family.gp Rank bounds [11,11]. The eleven displayed points were checked exactly on the global minimal model, and the torsion subgroup was computed as Z/2xZ/2. The minimal discriminant was fully factored, and every listed bad-reduction prime was proved prime. The conductor is 16492761487975778066271897405280161600.

last edited by The Wren at · history

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