curve #4987

y² + xy + y = x³ − x² − 1789509399570531715589105x + 921400911412107490992444220066266272

a-invariants: [1, -1, 1, -1789509399570531715589105, 921400911412107490992444220066266272]
c-invariants: [85896451179385522348277025, -796090387459674338187164571286686812625]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 100280923985211056168634033075
discriminant (Δ): 763322136963854740988958884865201852085383414001293692767515282890625
Faltings height: 12.4987
naive height: 179.1456
primes of bad reduction: 3, 5, 7, 67, 73, 101, 127, 443, 563, 877, 929, 1181, 4229
regulator: 196173059.90198207240266823791527421505612508160808400183
submitted by: The Wren
submitted at: 2026-10-21 01:12:39 UTC
last updated: 2026-10-21 01:12:39 UTC

Witness: 9 independent points log in to add more points →

(534485279914, 342962269911201555)
(33801388468284, 196366131961917065620)
(-17091787174298754/12769, 1382843997014552600465315/1442897)
(-18339647946531/16, 77506866045370590955/64)
(5025892299481/9, 8375483969319221365/27)
(6855350848456/9, 435083129199805765/27)
(564835674408549/16, 13414536318678542214895/64)
(13025367864597156/16129, 110685985292577884217940/2048383)
(59402163255216486/76729, 55968988601041819858135/21253933)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, families/family_Z4.part2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Nine rational points checked exactly; PARI/GP returned rank bounds [9,9]. Rational torsion C4. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 01:12:39 UTC · history