Elliptic curve search results

Conductor	j-invariant	Rank	Torsion	Complex multiplication
Bad$\ p$	Discriminant	Regulator	Analytic order of Ш	Galois image
Isogeny class size	Isogeny class degree	Cyclic isogeny degree	$p\ $div$\ $|Ш|	Nonmax$\ \ell$
Curves per isogeny class	Reduction	Integral points	Faltings height

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Results (1-50 of 302 matches)

 

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
11.a1	11a2	11.a	11a	$3$	$25$	\( 11 \)	\( -11 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$5$	25.120.0.3	5B.1.2	$1$	$1$		$0$	$5$	$0.496709$	$-52893159101157376/11$	$[0, -1, 1, -7820, -263580]$	\(y^2+y=x^3-x^2-7820x-263580\)
11.a2	11a1	11.a	11a	$3$	$25$	\( 11 \)	\( - 11^{5} \)	$0$	$\Z/5\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$5$	5.120.0.1	5Cs.1.1	$1$	$1$		$4$	$1$	$-0.308010$	$-122023936/161051$	$[0, -1, 1, -10, -20]$	\(y^2+y=x^3-x^2-10x-20\)
11.a3	11a3	11.a	11a	$3$	$25$	\( 11 \)	\( -11 \)	$0$	$\Z/5\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$5$	25.120.0.1	5B.1.1	$1$	$1$		$4$	$5$	$-1.112728$	$-4096/11$	$[0, -1, 1, 0, 0]$	\(y^2+y=x^3-x^2\)
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	$1$	$1$		$1$	$3$	$0.066527$	$-548347731625/1835008$	$[1, 0, 1, -171, -874]$	\(y^2+xy+y=x^3-171x-874\)
14.a3	14a2	14.a	14a	$6$	$18$	\( 2 \cdot 7 \)	\( 2^{3} \cdot 7^{6} \)	$0$	$\Z/6\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.6, 3.24.0.1	2B, 3Cs.1.1	$1$	$1$		$4$	$2$	$-0.136205$	$4956477625/941192$	$[1, 0, 1, -36, -70]$	\(y^2+xy+y=x^3-36x-70\)
14.a4	14a6	14.a	14a	$6$	$18$	\( 2 \cdot 7 \)	\( 2 \cdot 7^{2} \)	$0$	$\Z/6\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.6, 9.24.0.1	2B, 3B.1.1	$1$	$1$		$4$	$6$	$-0.685512$	$128787625/98$	$[1, 0, 1, -11, 12]$	\(y^2+xy+y=x^3-11x+12\)
14.a5	14a4	14.a	14a	$6$	$18$	\( 2 \cdot 7 \)	\( - 2^{2} \cdot 7 \)	$0$	$\Z/6\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.1, 9.24.0.1	2B, 3B.1.1	$1$	$1$		$5$	$3$	$-1.032085$	$-15625/28$	$[1, 0, 1, -1, 0]$	\(y^2+xy+y=x^3-x\)
14.a6	14a1	14.a	14a	$6$	$18$	\( 2 \cdot 7 \)	\( - 2^{6} \cdot 7^{3} \)	$0$	$\Z/6\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2, 3$	8.6.0.1, 3.24.0.1	2B, 3Cs.1.1	$1$	$1$		$5$	$1$	$-0.482779$	$9938375/21952$	$[1, 0, 1, 4, -6]$	\(y^2+xy+y=x^3+4x-6\)
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.a2	15a2	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( 3^{8} \cdot 5^{2} \)	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.96.0.58	2Cs	$1$	$1$		$2$	$2$	$-0.055704$	$272223782641/164025$	$[1, 1, 1, -135, -660]$	\(y^2+xy+y=x^3+x^2-135x-660\)
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	$1$	$1$		$0$	$4$	$0.290870$	$-147281603041/215233605$	$[1, 1, 1, -110, -880]$	\(y^2+xy+y=x^3+x^2-110x-880\)
15.a4	15a7	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( 3 \cdot 5 \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	32.96.0.5	2B	$1$	$1$		$2$	$4$	$-0.402277$	$56667352321/15$	$[1, 1, 1, -80, 242]$	\(y^2+xy+y=x^3+x^2-80x+242\)
15.a5	15a1	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( 3^{4} \cdot 5^{4} \)	$0$	$\Z/2\Z\oplus\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.96.0.3	2Cs	$1$	$1$		$6$	$1$	$-0.402277$	$111284641/50625$	$[1, 1, 1, -10, -10]$	\(y^2+xy+y=x^3+x^2-10x-10\)
15.a6	15a3	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( 3^{2} \cdot 5^{2} \)	$0$	$\Z/2\Z\oplus\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.96.0.7	2Cs	$1$	$1$		$6$	$2$	$-0.748851$	$13997521/225$	$[1, 1, 1, -5, 2]$	\(y^2+xy+y=x^3+x^2-5x+2\)
15.a7	15a8	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( - 3 \cdot 5 \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	32.96.0.5	2B	$1$	$1$		$3$	$4$	$-1.095425$	$-1/15$	$[1, 1, 1, 0, 0]$	\(y^2+xy+y=x^3+x^2\)
15.a8	15a4	15.a	15a	$8$	$16$	\( 3 \cdot 5 \)	\( - 3^{2} \cdot 5^{8} \)	$0$	$\Z/8\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.96.0.99	2B	$1$	$1$		$6$	$2$	$-0.055704$	$4733169839/3515625$	$[1, 1, 1, 35, -28]$	\(y^2+xy+y=x^3+x^2+35x-28\)
17.a1	17a3	17.a	17a	$4$	$4$	\( 17 \)	\( 17 \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	32.96.0.18	2B	$1$	$1$		$0$	$4$	$-0.376636$	$82483294977/17$	$[1, -1, 1, -91, -310]$	\(y^2+xy+y=x^3-x^2-91x-310\)
17.a2	17a2	17.a	17a	$4$	$4$	\( 17 \)	\( 17^{2} \)	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.96.0.13	2Cs	$1$	$1$		$2$	$2$	$-0.723209$	$20346417/289$	$[1, -1, 1, -6, -4]$	\(y^2+xy+y=x^3-x^2-6x-4\)
17.a3	17a1	17.a	17a	$4$	$4$	\( 17 \)	\( - 17^{4} \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.96.0.32	2B	$1$	$1$		$2$	$1$	$-0.376636$	$-35937/83521$	$[1, -1, 1, -1, -14]$	\(y^2+xy+y=x^3-x^2-x-14\)
17.a4	17a4	17.a	17a	$4$	$4$	\( 17 \)	\( 17 \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	32.96.0.10	2B	$1$	$1$		$3$	$4$	$-1.069784$	$35937/17$	$[1, -1, 1, -1, 0]$	\(y^2+xy+y=x^3-x^2-x\)
19.a1	19a2	19.a	19a	$3$	$9$	\( 19 \)	\( -19 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	27.72.0.2	3B.1.2	$1$	$1$		$0$	$3$	$0.033439$	$-50357871050752/19$	$[0, 1, 1, -769, -8470]$	\(y^2+y=x^3+x^2-769x-8470\)
19.a2	19a1	19.a	19a	$3$	$9$	\( 19 \)	\( - 19^{3} \)	$0$	$\Z/3\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	9.72.0.3	3Cs.1.1	$1$	$1$		$2$	$1$	$-0.515867$	$-89915392/6859$	$[0, 1, 1, -9, -15]$	\(y^2+y=x^3+x^2-9x-15\)
19.a3	19a3	19.a	19a	$3$	$9$	\( 19 \)	\( -19 \)	$0$	$\Z/3\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	27.72.0.1	3B.1.1	$1$	$1$		$2$	$3$	$-1.065172$	$32768/19$	$[0, 1, 1, 1, 0]$	\(y^2+y=x^3+x^2+x\)
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	$1$	$1$		$1$	$3$	$-0.034070$	$-20720464/15625$	$[0, 1, 0, -36, -140]$	\(y^2=x^3+x^2-36x-140\)
20.a3	20a2	20.a	20a	$4$	$6$	\( 2^{2} \cdot 5 \)	\( 2^{4} \cdot 5 \)	$0$	$\Z/6\Z$	$\Q$		$\mathrm{SU}(2)$			$2, 3$	8.12.0.22, 3.8.0.1	2B, 3B.1.1	$1$	$1$		$5$	$2$	$-0.929950$	$16384/5$	$[0, 1, 0, -1, 0]$	\(y^2=x^3+x^2-x\)
20.a4	20a1	20.a	20a	$4$	$6$	\( 2^{2} \cdot 5 \)	\( - 2^{8} \cdot 5^{2} \)	$0$	$\Z/6\Z$	$\Q$		$\mathrm{SU}(2)$			$2, 3$	8.12.0.37, 3.8.0.1	2B, 3B.1.1	$1$	$1$		$5$	$1$	$-0.583377$	$21296/25$	$[0, 1, 0, 4, 4]$	\(y^2=x^3+x^2+4x+4\)
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.a2	21a2	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3^{2} \cdot 7^{4} \)	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.34	2Cs	$1$	$1$		$2$	$2$	$-0.272368$	$13027640977/21609$	$[1, 0, 0, -49, -136]$	\(y^2+xy=x^3-49x-136\)
21.a3	21a3	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3^{8} \cdot 7 \)	$0$	$\Z/8\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.159	2B	$1$	$1$		$6$	$2$	$-0.272368$	$6570725617/45927$	$[1, 0, 0, -39, 90]$	\(y^2+xy=x^3-39x+90\)
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	$1$	$1$		$0$	$4$	$0.074205$	$-4354703137/17294403$	$[1, 0, 0, -34, -217]$	\(y^2+xy=x^3-34x-217\)
21.a5	21a1	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( 3^{4} \cdot 7^{2} \)	$0$	$\Z/2\Z\oplus\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	8.48.0.24	2Cs	$1$	$1$		$6$	$1$	$-0.618942$	$7189057/3969$	$[1, 0, 0, -4, -1]$	\(y^2+xy=x^3-4x-1\)
21.a6	21a4	21.a	21a	$6$	$8$	\( 3 \cdot 7 \)	\( - 3^{2} \cdot 7 \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	16.48.0.27	2B	$1$	$1$		$3$	$2$	$-0.965515$	$103823/63$	$[1, 0, 0, 1, 0]$	\(y^2+xy=x^3+x\)
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.a2	24a3	24.a	24a	$6$	$8$	\( 2^{3} \cdot 3 \)	\( 2^{10} \cdot 3 \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	16.96.0.106	2B	$1$	$1$		$3$	$2$	$-0.298779$	$28756228/3$	$[0, -1, 0, -64, 220]$	\(y^2=x^3-x^2-64x+220\)
24.a3	24a2	24.a	24a	$6$	$8$	\( 2^{3} \cdot 3 \)	\( 2^{10} \cdot 3^{4} \)	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.96.0.49	2Cs	$1$	$1$		$3$	$2$	$-0.298779$	$1556068/81$	$[0, -1, 0, -24, -36]$	\(y^2=x^3-x^2-24x-36\)
24.a4	24a1	24.a	24a	$6$	$8$	\( 2^{3} \cdot 3 \)	\( 2^{8} \cdot 3^{2} \)	$0$	$\Z/2\Z\oplus\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.96.0.42	2Cs	$1$	$1$		$7$	$1$	$-0.645352$	$35152/9$	$[0, -1, 0, -4, 4]$	\(y^2=x^3-x^2-4x+4\)
24.a5	24a4	24.a	24a	$6$	$8$	\( 2^{3} \cdot 3 \)	\( - 2^{4} \cdot 3 \)	$0$	$\Z/4\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	16.96.0.110	2B	$1$	$1$		$3$	$2$	$-0.991926$	$2048/3$	$[0, -1, 0, 1, 0]$	\(y^2=x^3-x^2+x\)
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	$1$	$1$		$1$	$4$	$0.047795$	$207646/6561$	$[0, -1, 0, 16, -180]$	\(y^2=x^3-x^2+16x-180\)
26.a1	26a2	26.a	26a	$3$	$9$	\( 2 \cdot 13 \)	\( - 2^{9} \cdot 13 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	9.24.0.3	3B.1.2	$1$	$1$		$0$	$6$	$0.054386$	$-10730978619193/6656$	$[1, 0, 1, -460, -3830]$	\(y^2+xy+y=x^3-460x-3830\)
26.a2	26a1	26.a	26a	$3$	$9$	\( 2 \cdot 13 \)	\( - 2^{3} \cdot 13^{3} \)	$0$	$\Z/3\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	3.24.0.1	3Cs.1.1	$1$	$1$		$2$	$2$	$-0.494920$	$-10218313/17576$	$[1, 0, 1, -5, -8]$	\(y^2+xy+y=x^3-5x-8\)
26.a3	26a3	26.a	26a	$3$	$9$	\( 2 \cdot 13 \)	\( - 2 \cdot 13 \)	$0$	$\Z/3\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	9.24.0.1	3B.1.1	$1$	$1$		$2$	$6$	$-1.044226$	$12167/26$	$[1, 0, 1, 0, 0]$	\(y^2+xy+y=x^3\)
26.b1	26b2	26.b	26b	$2$	$7$	\( 2 \cdot 13 \)	\( - 2 \cdot 13^{7} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$7$	7.48.0.5	7B.1.3	$1$	$1$		$0$	$14$	$0.289794$	$-1064019559329/125497034$	$[1, -1, 1, -213, -1257]$	\(y^2+xy+y=x^3-x^2-213x-1257\)
26.b2	26b1	26.b	26b	$2$	$7$	\( 2 \cdot 13 \)	\( - 2^{7} \cdot 13 \)	$0$	$\Z/7\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$7$	7.48.0.1	7B.1.1	$1$	$1$		$6$	$2$	$-0.683161$	$-2146689/1664$	$[1, -1, 1, -3, 3]$	\(y^2+xy+y=x^3-x^2-3x+3\)
27.a1	27a2	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{11} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-3})$	$-27$	$N(\mathrm{U}(1))$		✓	$3$	27.648.13.34	3B.1.2	$1$	$1$		$0$	$3$	$0.052148$	$-12288000$	$[0, 0, 1, -270, -1708]$	\(y^2+y=x^3-270x-1708\)
27.a2	27a4	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{5} \)	$0$	$\Z/3\Z$	$\Q(\sqrt{-3})$	$-27$	$N(\mathrm{U}(1))$		✓	$3$	27.648.13.25	3B.1.1	$1$	$1$		$2$	$9$	$-0.497158$	$-12288000$	$[0, 0, 1, -30, 63]$	\(y^2+y=x^3-30x+63\)
27.a3	27a1	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{9} \)	$0$	$\Z/3\Z$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$3$	27.1944.55.37	3Cs.1.1	$1$	$1$		$2$	$1$	$-0.497158$	$0$	$[0, 0, 1, 0, -7]$	\(y^2+y=x^3-7\)
27.a4	27a3	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{3} \)	$0$	$\Z/3\Z$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$3$	27.1944.55.31	3Cs.1.1	$1$	$1$		$2$	$3$	$-1.046465$	$0$	$[0, 0, 1, 0, 0]$	\(y^2+y=x^3\)
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	$1$	$1$		$0$	$24$	$0.564166$	$16778985534208729/81000$	$[1, 0, 1, -5334, -150368]$	\(y^2+xy+y=x^3-5334x-150368\)