curve #6120

y² + xy + y = x³ + x² − 10958572218642024278780x + 441782147883394334171891116337501

a-invariants: [1, 1, 1, -10958572218642024278780, 441782147883394334171891116337501]
c-invariants: [526011466494817165381441, -381699775775197790723225053255961825]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 1403350426911985538403637610899460834214448220801709234
discriminant (Δ): -88903418749089166432445734717963639955078242223752337359551725568
Faltings height: 11.4056
naive height: 163.8599
primes of bad reduction: 2, 3, 29, 41, 271, 152599, 383219, 12412664421137194517022733363095338101
regulator: 8575367206665900841099020.037089947601385478868181437442381646434732287161243
submitted by: Michael Rubinstein
submitted at: 2026-10-29 03:13:59 UTC
last updated: 2026-10-29 03:13:59 UTC

Witness: 19 independent points log in to add more points →

(-28357747469523582417227/245060641296, 1536281762524663312437694839349739/121313839624606656)
(368220653948549/6241, 383559140002229199791/493039)
(18210115034715893/299209, 84301581995509879134467/163667323)
(22437090456217253/32761, 3323698417728949678732549/5929741)
(79273511405260587/1394761, 2624676334834030493439593/1647212741)
(502072268202487421/1874161, 331878003079611223556945569/2565726409)
(1078946195449844653/11744329, 583804233097548784252335899/40247815483)
(4962924100987418229/91145209, 2221852536783730060763524411/870163310323)
(17059582032256804885/57320041, 66586747924585486548845467987/433970030411)
(254273850687663745485/4206601, 4054635992199165491761548489947/8627738651)
(746273666216948427117/8484436321, 9835329848054721784673968637399/781509913963631)
(12231260456008587511569/209749344256, 98068244030567819235908907218311/96061843679739904)
(6833107484183966686541/110952279025, 25418674662879136587923200525151/36957649381832375)
(72401963782465642620229/1171791095049, 953560490394229628879854385592157/1268455657852877157)
(96362292514162262859709/1508045400625, 2895663557276311664005433655365377/1851917453102515625)
(147190555159309561059157/173458089289, 56059458798825927178177158136650403/72242345401350587)
(4039812493355488942176333/88206339263329, 4967276655003487997371361242228084343/828418325839136358767)
(235165257867372865142658765/8174916890071009, 9062950671428679204738034216407578432951/739137129820242126274223)
(3323381603472582456595770421/60480721363041225, 34981255826156917588550134433120183369789/14873920080375590848549125)

Commentary

Mestre ten-root construction, roots 0,1,5,23,24,45,76,86,106,114: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 19, proved: 19 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 03:13:59 UTC · history