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curve #5973

y2 = x3 + x2 − 352213191570255081617330433x + 2536073917002054435490550943826454017263
a-invariants
[0, 1, 0, -352213191570255081617330433, 2536073917002054435490550943826454017263]
c-invariants
[16906233195372243917631860800, -2191167864289876469663008248929562062080000]
rank (lower bound)
≥ 20
torsion subgroup
trivial
conductor (N)
14574758999340624880130810214478793247772364291268217656405600
discriminant (Δ)
17902285514729447423945876678033270382912753678286046875972770042678170304000000
Faltings height
14.0672
naive height
194.9924
primes of bad reduction
2, 3, 5, 11, 19, 37, 41, 61, 42416743, 159382173259, 27491510819347, 1689478886988284087
regulator
520178938270898051.940599381645831230685934234104915164492674337545908865914542
submitted by
apkorte
submitted at
last updated

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Commentary

Specialization at T = 4/19 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -62238598979262328847367T^8 + 114310404580625317891332T^7 + 345964098754055988215811T^6 - 432486848349735134109966T^5 + 175844415801606548060475T^4 - 539957014758046237778784T^3 - 3466666840492299593283363T^2 + 1375801562449141429517514T - 457847843272340951907252, A6 = 6231113960947468288851293055208170T^12 - 17986527908278999930497783451836564T^11 - 63510145446922339309436221943522517T^10 + 196072827486310127387111443914446460T^9 + 329896017984738220842162864843822384T^8 - 977730912392624912334102291243654336T^7 - 1090216443601797579432503967842970390T^6 + 2401580225827262855577615397568975952T^5 + 1437988402318580839377591372944875134T^4 - 1577869275593930979134666639990964780T^3 + 1723891172774355272317470092631767475T^2 - 538001952495578595025088868536038092T + 119515751079584041083046899484291104. It is a 2-neighbour of the fibration given in curve#4187 (itself a 2-neighbour of the fibration in curve#3900): its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at · history

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