curve #3155

y² + xy + y = x³ − 440474450756994110470755811415987126x + 112359700106827942022431270950796422996711199577222648

a-invariants: [1, 0, 1, -440474450756994110470755811415987126, 112359700106827942022431270950796422996711199577222648]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2031307482740605522947279673767859802550
discriminant (Δ): 15559573661445620009429257907757165580663973144863822933798261071936780406781952645009825869208883256250000
Faltings height: 19.2726
naive height: 257.8331
primes of bad reduction: 2, 3, 5, 7, 13, 41, 149, 431, 449, 1069, 2447, 2579, 2687, 17209, 18743, 21529
regulator: 479900236198.360791655534746541727886110226954050411516159280
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:57 UTC
last updated: 2026-09-26 20:39:57 UTC

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

(-83834698092714168, -385613188183090474813206929)
(401385324022728537, 15065891830395937652029981)
(868269806925678012, 620073456836630055811410931)
(405323867587784637, 20359561954665515268295681)
(394921523865119673, 91491516624926721794857)
(324408915685454637, 60058527517639096320574681)
(78666748193455428297/121, 422560149937742892038202881651/1331)
(312484280266626033897/3481, 55696097314215749153564988126599/205379)
(410238599647675881, 26485355885556302980359901)
(-1712444706383958194507043/3083536, -2333342889415351179295310121678598979/5414689216)
(25523416744002046908417/52441, 1383419659065873235093739250155509/12008989)

Commentary

Dujella - Peral (2017). Source: https://dujella.github.io/tors/z2z2old1011121314.html (curve entry 20). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:39:57 UTC · history