Elliptic curve search results

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Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	CM	Sato-Tate	Semistable	Potentially good	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Regulator	$Ш_{\textrm{an}}$	Ш primes	Integral points	Modular degree	Faltings height	j-invariant	Weierstrass coefficients	Weierstrass equation
14.a1	14a5	14.a	14a	$6$	$18$	\( 2 \cdot 7 \)	\( 2^{9} \cdot 7^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.6, 9.24.0.3	2B, 3B.1.2	$1$	$1$		$0$	$6$	$0.413101$	$2251439055699625/25088$	$[1, 0, 1, -2731, -55146]$	\(y^2+xy+y=x^3-2731x-55146\)
14.a2	14a3	14.a	14a	$6$	$18$	\( 2 \cdot 7 \)	\( - 2^{18} \cdot 7 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.1, 9.24.0.3	2B, 3B.1.2	$1$	$1$		$1$	$3$	$0.066527$	$-548347731625/1835008$	$[1, 0, 1, -171, -874]$	\(y^2+xy+y=x^3-171x-874\)
15.a1	15a5	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( 3^{4} \cdot 5 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.96.0.148	2B	$1$	$1$		$0$	$4$	$0.290870$	$1114544804970241/405$	$[1, 1, 1, -2160, -39540]$	\(y^2+xy+y=x^3+x^2-2160x-39540\)
15.a3	15a6	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( - 3^{16} \cdot 5 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.96.0.153	2B	$1$	$1$		$0$	$4$	$0.290870$	$-147281603041/215233605$	$[1, 1, 1, -110, -880]$	\(y^2+xy+y=x^3+x^2-110x-880\)
17.a1	17a3	17.a	17a	$4$	$4$	\( 17 \)	\( 17 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	32.96.0.18	2B	$1$	$1$		$0$	$4$	$-0.376636$	$82483294977/17$	$[1, -1, 1, -91, -310]$	\(y^2+xy+y=x^3-x^2-91x-310\)
20.a1	20a4	20.a	20a	$4$	$6$	\( 2^{2} \cdot 5 \)	\( 2^{4} \cdot 5^{3} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2, 3$	8.12.0.22, 3.8.0.2	2B, 3B.1.2	$1$	$1$		$1$	$6$	$-0.380644$	$488095744/125$	$[0, 1, 0, -41, -116]$	\(y^2=x^3+x^2-41x-116\)
20.a2	20a3	20.a	20a	$4$	$6$	\( 2^{2} \cdot 5 \)	\( - 2^{8} \cdot 5^{6} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2, 3$	8.12.0.37, 3.8.0.2	2B, 3B.1.2	$1$	$1$		$1$	$3$	$-0.034070$	$-20720464/15625$	$[0, 1, 0, -36, -140]$	\(y^2=x^3+x^2-36x-140\)
21.a1	21a5	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3 \cdot 7^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.48.0.48	2B	$1$	$1$		$0$	$4$	$0.074205$	$53297461115137/147$	$[1, 0, 0, -784, -8515]$	\(y^2+xy=x^3-784x-8515\)
21.a4	21a6	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( - 3 \cdot 7^{8} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.177	2B	$1$	$1$		$0$	$4$	$0.074205$	$-4354703137/17294403$	$[1, 0, 0, -34, -217]$	\(y^2+xy=x^3-34x-217\)
24.a1	24a5	24.a	24a	$6$	$8$	\( 2^{3} \cdot 3 \)	\( 2^{11} \cdot 3^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	16.96.0.135	2B	$1$	$1$		$1$	$4$	$0.047795$	$3065617154/9$	$[0, -1, 0, -384, -2772]$	\(y^2=x^3-x^2-384x-2772\)
24.a6	24a6	24.a	24a	$6$	$8$	\( 2^{3} \cdot 3 \)	\( - 2^{11} \cdot 3^{8} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.96.0.166	2B	$1$	$1$		$1$	$4$	$0.047795$	$207646/6561$	$[0, -1, 0, 16, -180]$	\(y^2=x^3-x^2+16x-180\)
30.a1	30a7	30.a	30a	$8$	$12$	\( 2 \cdot 3 \cdot 5 \)	\( 2^{3} \cdot 3^{4} \cdot 5^{3} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.12.0.7, 3.8.0.2	2B, 3B.1.2	$1$	$1$		$0$	$24$	$0.564166$	$16778985534208729/81000$	$[1, 0, 1, -5334, -150368]$	\(y^2+xy+y=x^3-5334x-150368\)
30.a2	30a8	30.a	30a	$8$	$12$	\( 2 \cdot 3 \cdot 5 \)	\( 2^{3} \cdot 3 \cdot 5^{12} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.12.0.8, 3.8.0.2	2B, 3B.1.2	$1$	$1$		$0$	$24$	$0.564166$	$10316097499609/5859375000$	$[1, 0, 1, -454, -544]$	\(y^2+xy+y=x^3-454x-544\)
30.a7	30a3	30.a	30a	$8$	$12$	\( 2 \cdot 3 \cdot 5 \)	\( - 2^{12} \cdot 3 \cdot 5^{3} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.12.0.12, 3.8.0.2	2B, 3B.1.2	$1$	$1$		$1$	$6$	$-0.128981$	$-273359449/1536000$	$[1, 0, 1, -14, -64]$	\(y^2+xy+y=x^3-14x-64\)
32.a1	32a3	32.a	32a	$4$	$4$	\( 2^{5} \)	\( 2^{9} \)	$0$	$\Z/2\Z$	$\Q(\sqrt{-1})$	$-16$	$N(\mathrm{U}(1))$		✓	$2$	16.192.3.554	2B	$1$	$1$		$1$	$4$	$-0.617386$	$287496$	$[0, 0, 0, -11, -14]$	\(y^2=x^3-11x-14\)
33.a3	33a2	33.a	33a	$4$	$4$	\( 3 \cdot 11 \)	\( 3^{3} \cdot 11 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.12.0.6	2B	$1$	$1$		$1$	$6$	$-0.705779$	$30664297/297$	$[1, 1, 0, -6, -9]$	\(y^2+xy=x^3+x^2-6x-9\)
33.a4	33a4	33.a	33a	$4$	$4$	\( 3 \cdot 11 \)	\( - 3^{12} \cdot 11 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	4.12.0.8	2B	$1$	$1$		$0$	$6$	$-0.012632$	$9090072503/5845851$	$[1, 1, 0, 44, 55]$	\(y^2+xy=x^3+x^2+44x+55\)
34.a1	34a4	34.a	34a	$4$	$6$	\( 2 \cdot 17 \)	\( 2 \cdot 17^{6} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.6, 3.8.0.2	2B, 3B.1.2	$1$	$1$		$0$	$12$	$0.179487$	$159661140625/48275138$	$[1, 0, 0, -113, -329]$	\(y^2+xy=x^3-113x-329\)
34.a2	34a3	34.a	34a	$4$	$6$	\( 2 \cdot 17 \)	\( 2^{2} \cdot 17^{3} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.1, 3.8.0.2	2B, 3B.1.2	$1$	$1$		$1$	$6$	$-0.167086$	$120920208625/19652$	$[1, 0, 0, -103, -411]$	\(y^2+xy=x^3-103x-411\)
36.a1	36a4	36.a	36a	$4$	$6$	\( 2^{2} \cdot 3^{2} \)	\( 2^{8} \cdot 3^{9} \)	$0$	$\Z/2\Z$	$\Q(\sqrt{-3})$	$-12$	$N(\mathrm{U}(1))$		✓	$3$	27.648.18.4	3B.1.2	$1$	$1$		$1$	$6$	$0.080464$	$54000$	$[0, 0, 0, -135, -594]$	\(y^2=x^3-135x-594\)
36.a3	36a3	36.a	36a	$4$	$6$	\( 2^{2} \cdot 3^{2} \)	\( - 2^{4} \cdot 3^{9} \)	$0$	$\Z/2\Z$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$2, 3$	16.192.9.83, 27.648.18.4	2B, 3B.1.2	$1$	$1$		$1$	$3$	$-0.266109$	$0$	$[0, 0, 0, 0, -27]$	\(y^2=x^3-27\)
39.a1	39a2	39.a	39a	$4$	$4$	\( 3 \cdot 13 \)	\( 3^{4} \cdot 13 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	4.12.0.8	2B	$1$	$1$		$0$	$4$	$-0.318367$	$37159393753/1053$	$[1, 1, 0, -69, -252]$	\(y^2+xy=x^3+x^2-69x-252\)
39.a4	39a4	39.a	39a	$4$	$4$	\( 3 \cdot 13 \)	\( - 3 \cdot 13 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.12.0.6	2B	$1$	$1$		$1$	$4$	$-1.011513$	$12167/39$	$[1, 1, 0, 1, 0]$	\(y^2+xy=x^3+x^2+x\)
40.a1	40a2	40.a	40a	$4$	$4$	\( 2^{3} \cdot 5 \)	\( 2^{10} \cdot 5 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	16.48.0.55	2B	$1$	$1$		$1$	$4$	$-0.202214$	$132304644/5$	$[0, 0, 0, -107, -426]$	\(y^2=x^3-107x-426\)
42.a2	42a5	42.a	42a	$6$	$8$	\( 2 \cdot 3 \cdot 7 \)	\( 2 \cdot 3^{4} \cdot 7^{8} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.96.0.164	2B	$1$	$1$		$0$	$32$	$0.535675$	$84448510979617/933897762$	$[1, 1, 1, -914, -10915]$	\(y^2+xy+y=x^3+x^2-914x-10915\)
42.a6	42a6	42.a	42a	$6$	$8$	\( 2 \cdot 3 \cdot 7 \)	\( - 2 \cdot 3^{16} \cdot 7^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.96.0.137	2B	$1$	$1$		$0$	$32$	$0.535675$	$6359387729183/4218578658$	$[1, 1, 1, 386, 1277]$	\(y^2+xy+y=x^3+x^2+386x+1277\)
45.a1	45a7	45.a	45a	$8$	$16$	\( 3^{2} \cdot 5 \)	\( 3^{10} \cdot 5 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	32.96.0.51	2B	$1$	$1$		$0$	$32$	$0.840176$	$1114544804970241/405$	$[1, -1, 0, -19440, 1048135]$	\(y^2+xy=x^3-x^2-19440x+1048135\)
45.a3	45a8	45.a	45a	$8$	$16$	\( 3^{2} \cdot 5 \)	\( - 3^{22} \cdot 5 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	32.96.0.67	2B	$1$	$1$		$0$	$32$	$0.840176$	$-147281603041/215233605$	$[1, -1, 0, -990, 22765]$	\(y^2+xy=x^3-x^2-990x+22765\)
45.a4	45a3	45.a	45a	$8$	$16$	\( 3^{2} \cdot 5 \)	\( 3^{7} \cdot 5 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	32.96.0.77	2B	$1$	$1$		$0$	$8$	$0.147029$	$56667352321/15$	$[1, -1, 0, -720, -7259]$	\(y^2+xy=x^3-x^2-720x-7259\)
45.a7	45a1	45.a	45a	$8$	$16$	\( 3^{2} \cdot 5 \)	\( - 3^{7} \cdot 5 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	32.96.0.31	2B	$1$	$1$		$1$	$2$	$-0.546119$	$-1/15$	$[1, -1, 0, 0, -5]$	\(y^2+xy=x^3-x^2-5\)
45.a8	45a6	45.a	45a	$8$	$16$	\( 3^{2} \cdot 5 \)	\( - 3^{8} \cdot 5^{8} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.96.0.172	2B	$1$	$1$		$0$	$16$	$0.493602$	$4733169839/3515625$	$[1, -1, 0, 315, 1066]$	\(y^2+xy=x^3-x^2+315x+1066\)
46.a1	46a2	46.a	46a	$2$	$2$	\( 2 \cdot 23 \)	\( 2^{5} \cdot 23^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.6.0.6	2B	$1$	$1$		$0$	$10$	$-0.091908$	$545138290809/16928$	$[1, -1, 0, -170, -812]$	\(y^2+xy=x^3-x^2-170x-812\)
46.a2	46a1	46.a	46a	$2$	$2$	\( 2 \cdot 23 \)	\( - 2^{10} \cdot 23 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.6.0.1	2B	$1$	$1$		$1$	$5$	$-0.438482$	$-116930169/23552$	$[1, -1, 0, -10, -12]$	\(y^2+xy=x^3-x^2-10x-12\)
48.a2	48a2	48.a	48a	$6$	$8$	\( 2^{4} \cdot 3 \)	\( 2^{10} \cdot 3 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	16.96.0.150	2B	$1$	$1$		$1$	$4$	$-0.298779$	$28756228/3$	$[0, 1, 0, -64, -220]$	\(y^2=x^3+x^2-64x-220\)
48.a5	48a4	48.a	48a	$6$	$8$	\( 2^{4} \cdot 3 \)	\( - 2^{4} \cdot 3 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	16.96.0.142	2B	$1$	$1$		$1$	$4$	$-0.991926$	$2048/3$	$[0, 1, 0, 1, 0]$	\(y^2=x^3+x^2+x\)
49.a1	49a4	49.a	49a	$4$	$14$	\( 7^{2} \)	\( 7^{9} \)	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-28$	$N(\mathrm{U}(1))$		✓	$7$	7.48.0.3	7B.1.2	$1$	$1$		$0$	$14$	$0.520391$	$16581375$	$[1, -1, 0, -1822, 30393]$	\(y^2+xy=x^3-x^2-1822x+30393\)
49.a2	49a3	49.a	49a	$4$	$14$	\( 7^{2} \)	\( - 7^{9} \)	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-7$	$N(\mathrm{U}(1))$		✓	$7$	7.48.0.3	7B.1.2	$1$	$1$		$1$	$7$	$0.173817$	$-3375$	$[1, -1, 0, -107, 552]$	\(y^2+xy=x^3-x^2-107x+552\)
49.a3	49a2	49.a	49a	$4$	$14$	\( 7^{2} \)	\( 7^{3} \)	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-28$	$N(\mathrm{U}(1))$		✓	$7$	7.48.0.6	7B.1.5	$1$	$1$		$0$	$2$	$-0.452565$	$16581375$	$[1, -1, 0, -37, -78]$	\(y^2+xy=x^3-x^2-37x-78\)
49.a4	49a1	49.a	49a	$4$	$14$	\( 7^{2} \)	\( - 7^{3} \)	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-7$	$N(\mathrm{U}(1))$		✓	$7$	7.48.0.6	7B.1.5	$1$	$1$		$1$	$1$	$-0.799138$	$-3375$	$[1, -1, 0, -2, -1]$	\(y^2+xy=x^3-x^2-2x-1\)
52.a1	52a2	52.a	52a	$2$	$2$	\( 2^{2} \cdot 13 \)	\( 2^{4} \cdot 13 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.12.0.22	2B	$1$	$1$		$1$	$6$	$-0.777873$	$442368/13$	$[0, 0, 0, -4, -3]$	\(y^2=x^3-4x-3\)
52.a2	52a1	52.a	52a	$2$	$2$	\( 2^{2} \cdot 13 \)	\( - 2^{8} \cdot 13^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.12.0.37	2B	$1$	$1$		$1$	$3$	$-0.431299$	$432/169$	$[0, 0, 0, 1, -10]$	\(y^2=x^3+x-10\)
55.a2	55a2	55.a	55a	$4$	$4$	\( 5 \cdot 11 \)	\( 5 \cdot 11^{4} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	4.12.0.8	2B	$1$	$1$		$0$	$4$	$-0.285666$	$2749884201/73205$	$[1, -1, 0, -29, -52]$	\(y^2+xy=x^3-x^2-29x-52\)
55.a4	55a4	55.a	55a	$4$	$4$	\( 5 \cdot 11 \)	\( - 5 \cdot 11 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.12.0.6	2B	$1$	$1$		$1$	$4$	$-0.978812$	$59319/55$	$[1, -1, 0, 1, 0]$	\(y^2+xy=x^3-x^2+x\)
56.a1	56a4	56.a	56a	$4$	$4$	\( 2^{3} \cdot 7 \)	\( 2^{11} \cdot 7 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.24.0.103	2B	$1$	$1$		$1$	$8$	$-0.002033$	$1443468546/7$	$[0, 0, 0, -299, 1990]$	\(y^2=x^3-299x+1990\)
56.a2	56a3	56.a	56a	$4$	$4$	\( 2^{3} \cdot 7 \)	\( 2^{11} \cdot 7^{4} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.24.0.58	2B	$1$	$1$		$1$	$8$	$-0.002033$	$11090466/2401$	$[0, 0, 0, -59, -138]$	\(y^2=x^3-59x-138\)
56.b1	56b2	56.b	56b	$2$	$2$	\( 2^{3} \cdot 7 \)	\( 2^{11} \cdot 7^{2} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.6.0.6	2B	$1$	$1$		$1$	$8$	$-0.234669$	$3543122/49$	$[0, -1, 0, -40, -84]$	\(y^2=x^3-x^2-40x-84\)
56.b2	56b1	56.b	56b	$2$	$2$	\( 2^{3} \cdot 7 \)	\( - 2^{10} \cdot 7 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.6.0.1	2B	$1$	$1$		$1$	$4$	$-0.581243$	$-4/7$	$[0, -1, 0, 0, -4]$	\(y^2=x^3-x^2-4\)
57.c3	57b2	57.c	57b	$4$	$4$	\( 3 \cdot 19 \)	\( 3 \cdot 19 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.12.0.6	2B	$1$	$1$		$1$	$6$	$-0.938156$	$389017/57$	$[1, 0, 1, -2, -1]$	\(y^2+xy+y=x^3-2x-1\)
57.c4	57b4	57.c	57b	$4$	$4$	\( 3 \cdot 19 \)	\( - 3 \cdot 19^{4} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	4.12.0.8	2B	$1$	$1$		$0$	$6$	$-0.245009$	$67419143/390963$	$[1, 0, 1, 8, 29]$	\(y^2+xy+y=x^3+8x+29\)
62.a1	62a4	62.a	62a	$4$	$4$	\( 2 \cdot 31 \)	\( 2 \cdot 31 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.24.0.103	2B	$1$	$1$		$0$	$8$	$-0.108078$	$3999236143617/62$	$[1, -1, 1, -331, 2397]$	\(y^2+xy+y=x^3-x^2-331x+2397\)

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