Recent activity
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commented on curve #3571
Inspired by (much better) curve#3566 Specialized curve of the C2 x C4 family y^2 = x(x+u^2)(x+v^2), where u±v=d*w^2. Here (u,v)=(1652,9284), with v-u=7632=53*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4. -
submitted curve #3571 — rank ≥ 4, torsion ℤ/2ℤ × ℤ/4ℤ, log |Δ| = 88.80
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commented on curve #3570
Inspired by (much better) curve#3566 Specialized curve of the C2 x C4 family y^2 = x(x+u^2)(x+v^2), where u±v=d*w^2. Here (u,v)=(1270,7462), with v-u=6192=43*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4. -
submitted curve #3570 — rank ≥ 4, torsion ℤ/2ℤ × ℤ/4ℤ, log |Δ| = 94.32
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commented on curve #3569
Inspired by (much better) curve#3566 Specialized curve of the C2 x C4 family y^2 = x(x+u^2)(x+v^2), where u±v=d*w^2. Here (u,v)=(1180,5356), with v-u=4176=29*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4. -
submitted curve #3569 — rank ≥ 4, torsion ℤ/2ℤ × ℤ/4ℤ, log |Δ| = 83.02
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commented on curve #3568
Specialization at T = 236253/205754 of a rank-17 fibration of X1008 (2-neighbour of the one in curve#751). -
submitted curve #3568 — rank ≥ 30, log |Δ| = 393.30
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commented on curve #3567
Specialization at T = 82431/14059 of a rank-17 fibration of X1008 (2-neighbour of the one in curve#751). -
submitted curve #3567 — rank ≥ 30, log |Δ| = 393.62
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commented on curve #3566
Found in a targeted low-conductor search in the C2 x C4 family y^2 = x(x+u^2)(x+v^2), using low-squarefree-kernel loci u±v=d*w^2. Here (u,v)=(497,5825), with v-u=5328=37*12^2. PARI/GP gives exact rank 5 and torsion C2 x C4. -
submitted curve #3566 — rank ≥ 5, torsion ℤ/2ℤ × ℤ/4ℤ, log |Δ| = 96.95
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commented on curve #3565
2-isogenous to the specialization at T = -4834/987 of the rank-9 fibration of X400 (equations as in curve#802). -
submitted curve #3565 — rank ≥ 17, torsion ℤ/2ℤ, log |Δ| = 247.46 ☆ record for ℤ/2ℤ torsion, rank ≥ 17: conductor, discriminant, Faltings height, naive height
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commented on curve #3564
2-isogenous quotient of curve#3392 by the rational 2-torsion point (-74,0). Torsion Z/4Z; PARI/GP ellrank gives [4,4,0], so the rank is exactly 4. The four witnesses are the images of the independent witnesses of curve#3392 under the degree-2 isogeny. Naive height 54.7321239517, … -
submitted curve #3564 — rank ≥ 4, torsion ℤ/4ℤ, log |Δ| = 47.36
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commented on curve #3563
Specialization at T = -1/5 of a rank-17 fibration of X1008 (2-neighbour of the one in curve#751). -
submitted curve #3563 — rank ≥ 20, log |Δ| = 157.23 ★ record for rank ≥ 20: discriminant, Faltings height, naive height
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commented on curve #3557
[edit by 7fff-zip] Fibre T=-953/2683 of fibration ICF_3545 (reduced frame); points: sections + fake 2-descent (Elkies) + iterated bootstrap. Certified rank >= 29 by 2-descent images mod primes up to 163. X1092 fibration. -
commented on curve #3557
Fibre T=-953/2683 of fibration ICF_3545 (reduced frame); points: sections + fake 2-descent (Elkies) + iterated bootstrap. rank-32-003. Certified rank >= 29 by 2-descent images mod primes up to 163. X1092 fibration of ICARM #3545 (model data/icarm_fibrations/nf_3545.gp, sections r… -
recorded the primes of bad reduction of curve #3556 — rank ≥ 28, log |Δ| = 334.97
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commented on curve #3562
Fibre at t = 83497/251378 of the Z/4 fibration E1 of X163 from Elkies–Klagsbrun (arXiv:2003.00077, Sec. 9), who list this t among their rank-12 parameters. The 12 generators were computed with PARI ellrank (2-isogeny descent). -
submitted curve #3562 — rank ≥ 12, torsion ℤ/4ℤ, log |Δ| = 272.38 ☆ record for ℤ/4ℤ torsion, rank ≥ 12: discriminant, Faltings height, naive height
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commented on curve #3561
Specialization at T = 13601/15 of a rank-17 fibration of X1008 (2-neighbour of the one in curve#751). -
submitted curve #3561 — rank ≥ 28, log |Δ| = 315.46
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submitted curve #3560 — rank ≥ 27, log |Δ| = 278.45
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submitted curve #3559 — rank ≥ 5, torsion ℤ/8ℤ, log |Δ| = 196.96
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submitted curve #3558 — rank ≥ 5, torsion ℤ/8ℤ, log |Δ| = 189.77
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recorded the primes of bad reduction of curve #3545 — rank ≥ 31, log |Δ| = 458.63
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recorded the primes of bad reduction of curve #3557 — rank ≥ 29, log |Δ| = 346.58