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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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
389.a1	389a1	389.a	389a	$1$	$1$	\( 389 \)	\( 389 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.152460177$	$1$		$20$	$40$	$-0.795642$	$1404928/389$	$[0, 1, 1, -2, 0]$	\(y^2+y=x^3+x^2-2x\)
433.a1	433a1	433.a	433a	$1$	$1$	\( 433 \)	\( -433 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.224694163$	$1$		$16$	$28$	$-0.815220$	$-1/433$	$[1, 0, 0, 0, 1]$	\(y^2+xy=x^3+1\)
446.a1	446d1	446.a	446d	$1$	$1$	\( 2 \cdot 223 \)	\( 2^{2} \cdot 223 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.097343097$	$1$		$24$	$88$	$-0.700399$	$8120601/892$	$[1, -1, 0, -4, 4]$	\(y^2+xy=x^3-x^2-4x+4\)
563.a1	563a1	563.a	563a	$1$	$1$	\( 563 \)	\( -563 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.219229003$	$1$		$14$	$52$	$-0.571680$	$-374805361/563$	$[1, 1, 1, -15, 16]$	\(y^2+xy+y=x^3+x^2-15x+16\)
571.a1	571b1	571.a	571b	$1$	$1$	\( 571 \)	\( -571 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.177253140$	$1$		$16$	$48$	$-0.717502$	$-8998912/571$	$[0, 1, 1, -4, 2]$	\(y^2+y=x^3+x^2-4x+2\)
643.a1	643a1	643.a	643a	$1$	$1$	\( 643 \)	\( -643 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.229179621$	$1$		$14$	$32$	$-0.717735$	$-7189057/643$	$[1, 0, 0, -4, 3]$	\(y^2+xy=x^3-4x+3\)
655.a1	655a1	655.a	655a	$1$	$1$	\( 5 \cdot 131 \)	\( - 5^{2} \cdot 131 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.098953519$	$1$		$22$	$144$	$-0.518141$	$-242970624/3275$	$[0, 0, 1, -13, 18]$	\(y^2+y=x^3-13x+18\)
664.a1	664a1	664.a	664a	$1$	$1$	\( 2^{3} \cdot 83 \)	\( - 2^{8} \cdot 83 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.096676050$	$1$		$26$	$160$	$-0.466387$	$-148176/83$	$[0, 0, 0, -7, 10]$	\(y^2=x^3-7x+10\)
681.a1	681c1	681.a	681c	$1$	$1$	\( 3 \cdot 227 \)	\( - 3^{2} \cdot 227 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.136079688$	$1$		$22$	$96$	$-0.685717$	$-4096/2043$	$[0, -1, 1, 0, 2]$	\(y^2+y=x^3-x^2+2\)
707.a1	707a1	707.a	707a	$1$	$1$	\( 7 \cdot 101 \)	\( 7^{2} \cdot 101 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.112857456$	$1$		$22$	$104$	$-0.506146$	$207474688/4949$	$[0, 1, 1, -12, 12]$	\(y^2+y=x^3+x^2-12x+12\)
709.a1	709a1	709.a	709a	$1$	$1$	\( 709 \)	\( 709 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.258683172$	$1$		$16$	$44$	$-0.760408$	$1404928/709$	$[0, -1, 1, -2, 0]$	\(y^2+y=x^3-x^2-2x\)
718.a1	718b1	718.a	718b	$1$	$1$	\( 2 \cdot 359 \)	\( 2^{4} \cdot 359 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.169858359$	$1$		$20$	$112$	$-0.588549$	$10218313/5744$	$[1, 0, 1, -5, 0]$	\(y^2+xy+y=x^3-5x\)
794.a1	794a1	794.a	794a	$1$	$1$	\( 2 \cdot 397 \)	\( - 2^{2} \cdot 397 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.149049991$	$1$		$20$	$88$	$-0.688844$	$-1771561/1588$	$[1, 0, 1, -3, 2]$	\(y^2+xy+y=x^3-3x+2\)
817.a1	817a1	817.a	817a	$1$	$1$	\( 19 \cdot 43 \)	\( - 19^{2} \cdot 43 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.256256382$	$1$		$14$	$56$	$-0.516743$	$32768/15523$	$[0, 1, 1, 1, 6]$	\(y^2+y=x^3+x^2+x+6\)
916.a1	916c1	916.a	916c	$1$	$1$	\( 2^{2} \cdot 229 \)	\( 2^{4} \cdot 229 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.136468199$	$1$		$22$	$132$	$-0.624127$	$442368/229$	$[0, 0, 0, -4, 1]$	\(y^2=x^3-4x+1\)
944.a1	944e1	944.a	944e	$1$	$1$	\( 2^{4} \cdot 59 \)	\( - 2^{10} \cdot 59 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.117860680$	$1$		$24$	$224$	$-0.335824$	$-740772/59$	$[0, 0, 0, -19, 34]$	\(y^2=x^3-19x+34\)
997.a1	997c1	997.a	997c	$1$	$1$	\( 997 \)	\( 997 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.249705466$	$1$		$16$	$96$	$-0.482781$	$1593413632/997$	$[0, -1, 1, -24, 54]$	\(y^2+y=x^3-x^2-24x+54\)
997.c1	997b1	997.c	997b	$1$	$1$	\( 997 \)	\( 997 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.571019259$	$1$		$10$	$48$	$-0.671631$	$16777216/997$	$[0, -1, 1, -5, -3]$	\(y^2+y=x^3-x^2-5x-3\)
1001.a1	1001c1	1001.a	1001c	$1$	$1$	\( 7 \cdot 11 \cdot 13 \)	\( - 7^{2} \cdot 11^{3} \cdot 13^{2} \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.036010745$	$1$		$32$	$1008$	$0.161095$	$-871531204608/11022011$	$[0, 0, 1, -199, 1092]$	\(y^2+y=x^3-199x+1092\)
1028.a1	1028a1	1028.a	1028a	$1$	$1$	\( 2^{2} \cdot 257 \)	\( 2^{4} \cdot 257 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.136468233$	$1$		$20$	$132$	$-0.534093$	$7626496/257$	$[0, 1, 0, -10, 9]$	\(y^2=x^3+x^2-10x+9\)
1034.a1	1034a1	1034.a	1034a	$1$	$1$	\( 2 \cdot 11 \cdot 47 \)	\( - 2^{2} \cdot 11 \cdot 47 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.157767968$	$1$		$20$	$128$	$-0.552691$	$-169112377/2068$	$[1, 0, 1, -12, 14]$	\(y^2+xy+y=x^3-12x+14\)
1058.a1	1058c2	1058.a	1058c	$2$	$3$	\( 2 \cdot 23^{2} \)	\( - 2^{6} \cdot 23^{2} \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 3$	4.2.0.1, 3.4.0.1	3B	$0.206240920$	$1$		$20$	$240$	$-0.133320$	$-313994137/64$	$[1, 0, 1, -115, 462]$	\(y^2+xy+y=x^3-115x+462\)
1058.a2	1058c1	1058.a	1058c	$2$	$3$	\( 2 \cdot 23^{2} \)	\( - 2^{2} \cdot 23^{2} \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 3$	4.2.0.1, 3.4.0.1	3B	$0.206240920$	$1$		$16$	$80$	$-0.682627$	$23/4$	$[1, 0, 1, 0, 2]$	\(y^2+xy+y=x^3+2\)
1070.a1	1070a1	1070.a	1070a	$1$	$1$	\( 2 \cdot 5 \cdot 107 \)	\( - 2^{2} \cdot 5^{2} \cdot 107 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.087830480$	$1$		$26$	$240$	$-0.484061$	$-116930169/10700$	$[1, -1, 0, -10, 16]$	\(y^2+xy=x^3-x^2-10x+16\)
1073.a1	1073a1	1073.a	1073a	$1$	$1$	\( 29 \cdot 37 \)	\( 29^{2} \cdot 37 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.250120597$	$1$		$12$	$128$	$-0.268924$	$10303307776/31117$	$[0, -1, 1, -45, 132]$	\(y^2+y=x^3-x^2-45x+132\)
1077.a1	1077a1	1077.a	1077a	$1$	$1$	\( 3 \cdot 359 \)	\( 3^{4} \cdot 359 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.201754913$	$1$		$16$	$144$	$-0.336448$	$2181825073/29079$	$[1, 1, 1, -27, 42]$	\(y^2+xy+y=x^3+x^2-27x+42\)
1088.a1	1088j2	1088.a	1088j	$2$	$2$	\( 2^{6} \cdot 17 \)	\( 2^{15} \cdot 17^{2} \)	$2$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.6.0.6	2B	$0.531665631$	$1$		$25$	$384$	$0.043360$	$941192/289$	$[0, 1, 0, -65, -161]$	\(y^2=x^3+x^2-65x-161\)
1088.a2	1088j1	1088.a	1088j	$2$	$2$	\( 2^{6} \cdot 17 \)	\( 2^{12} \cdot 17 \)	$2$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.6.0.1	2B	$0.531665631$	$1$		$21$	$192$	$-0.303214$	$438976/17$	$[0, 1, 0, -25, 39]$	\(y^2=x^3+x^2-25x+39\)
1094.a1	1094a1	1094.a	1094a	$1$	$1$	\( 2 \cdot 547 \)	\( - 2^{2} \cdot 547 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.166108184$	$1$		$20$	$88$	$-0.609101$	$-30664297/2188$	$[1, 0, 1, -7, 6]$	\(y^2+xy+y=x^3-7x+6\)
1102.a1	1102a1	1102.a	1102a	$1$	$1$	\( 2 \cdot 19 \cdot 29 \)	\( - 2^{6} \cdot 19^{2} \cdot 29 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.148211786$	$1$		$18$	$288$	$-0.162938$	$-2845178713/670016$	$[1, 1, 0, -29, 61]$	\(y^2+xy=x^3+x^2-29x+61\)
1126.a1	1126a1	1126.a	1126a	$1$	$1$	\( 2 \cdot 563 \)	\( - 2^{4} \cdot 563 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.229392466$	$1$		$18$	$176$	$-0.560435$	$658503/9008$	$[1, -1, 0, 2, 4]$	\(y^2+xy=x^3-x^2+2x+4\)
1132.a1	1132a1	1132.a	1132a	$1$	$1$	\( 2^{2} \cdot 283 \)	\( - 2^{4} \cdot 283 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.150224255$	$1$		$20$	$120$	$-0.582228$	$-1048576/283$	$[0, 1, 0, -5, 4]$	\(y^2=x^3+x^2-5x+4\)
1137.a1	1137a1	1137.a	1137a	$1$	$1$	\( 3 \cdot 379 \)	\( - 3^{2} \cdot 379 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.198423842$	$1$		$16$	$72$	$-0.636436$	$-912673/3411$	$[1, 1, 1, -2, 2]$	\(y^2+xy+y=x^3+x^2-2x+2\)
1141.a1	1141a1	1141.a	1141a	$1$	$1$	\( 7 \cdot 163 \)	\( - 7^{5} \cdot 163 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.123412280$	$1$		$20$	$280$	$-0.071379$	$-2181825073/2739541$	$[1, 0, 0, -27, 94]$	\(y^2+xy=x^3-27x+94\)
1143.a1	1143c1	1143.a	1143c	$1$	$1$	\( 3^{2} \cdot 127 \)	\( 3^{7} \cdot 127 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.113423175$	$1$		$24$	$352$	$-0.191020$	$8998912/381$	$[0, 0, 1, -39, 90]$	\(y^2+y=x^3-39x+90\)
1147.b1	1147a1	1147.b	1147a	$1$	$1$	\( 31 \cdot 37 \)	\( 31^{2} \cdot 37 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.291623432$	$1$		$14$	$80$	$-0.428352$	$89915392/35557$	$[0, -1, 1, -9, 9]$	\(y^2+y=x^3-x^2-9x+9\)
1171.a1	1171a1	1171.a	1171a	$1$	$1$	\( 1171 \)	\( 1171 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.336702983$	$1$		$14$	$88$	$-0.720388$	$2146689/1171$	$[1, -1, 1, -3, 0]$	\(y^2+xy+y=x^3-x^2-3x\)
1246.a1	1246c1	1246.a	1246c	$1$	$1$	\( 2 \cdot 7 \cdot 89 \)	\( - 2^{4} \cdot 7^{2} \cdot 89 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.127061335$	$1$		$24$	$320$	$-0.391431$	$-185193/69776$	$[1, -1, 0, -1, 13]$	\(y^2+xy=x^3-x^2-x+13\)
1309.a1	1309b1	1309.a	1309b	$1$	$1$	\( 7 \cdot 11 \cdot 17 \)	\( - 7^{2} \cdot 11 \cdot 17^{2} \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.106196687$	$1$		$22$	$256$	$-0.269675$	$-1231925248/155771$	$[0, -1, 1, -22, 52]$	\(y^2+y=x^3-x^2-22x+52\)
1324.a1	1324a1	1324.a	1324a	$1$	$1$	\( 2^{2} \cdot 331 \)	\( - 2^{4} \cdot 331 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.222923629$	$1$		$18$	$108$	$-0.601645$	$131072/331$	$[0, 1, 0, 3, 4]$	\(y^2=x^3+x^2+3x+4\)
1325.a1	1325e1	1325.a	1325e	$1$	$1$	\( 5^{2} \cdot 53 \)	\( 5^{4} \cdot 53 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.143609349$	$1$		$22$	$216$	$-0.440645$	$102400/53$	$[0, 1, 1, -8, -6]$	\(y^2+y=x^3+x^2-8x-6\)
1431.a1	1431a1	1431.a	1431a	$1$	$1$	\( 3^{3} \cdot 53 \)	\( 3^{9} \cdot 53 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.286131263$	$1$		$16$	$288$	$-0.147027$	$132651/53$	$[1, -1, 1, -29, -26]$	\(y^2+xy+y=x^3-x^2-29x-26\)
1436.a1	1436a1	1436.a	1436a	$1$	$1$	\( 2^{2} \cdot 359 \)	\( 2^{8} \cdot 359 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.227823191$	$1$		$18$	$192$	$-0.351953$	$810448/359$	$[0, 1, 0, -12, 4]$	\(y^2=x^3+x^2-12x+4\)
1443.a1	1443c1	1443.a	1443c	$2$	$2$	\( 3 \cdot 13 \cdot 37 \)	\( 3^{2} \cdot 13 \cdot 37 \)	$2$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	2.3.0.1	2B	$0.790463101$	$1$		$15$	$112$	$-0.545658$	$81182737/4329$	$[1, 1, 1, -9, 6]$	\(y^2+xy+y=x^3+x^2-9x+6\)
1443.a2	1443c2	1443.a	1443c	$2$	$2$	\( 3 \cdot 13 \cdot 37 \)	\( - 3 \cdot 13^{2} \cdot 37^{2} \)	$2$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$2$	2.3.0.1	2B	$0.790463101$	$1$		$16$	$224$	$-0.199084$	$23639903/694083$	$[1, 1, 1, 6, 42]$	\(y^2+xy+y=x^3+x^2+6x+42\)
1446.a1	1446a1	1446.a	1446a	$1$	$1$	\( 2 \cdot 3 \cdot 241 \)	\( - 2^{2} \cdot 3^{2} \cdot 241 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.119198913$	$1$		$24$	$192$	$-0.546685$	$-10218313/8676$	$[1, 1, 0, -4, 4]$	\(y^2+xy=x^3+x^2-4x+4\)
1466.b1	1466b1	1466.b	1466b	$1$	$1$	\( 2 \cdot 733 \)	\( 2^{10} \cdot 733 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.088731563$	$1$		$30$	$1040$	$-0.134249$	$8012006001/750592$	$[1, -1, 1, -42, 105]$	\(y^2+xy+y=x^3-x^2-42x+105\)
1477.a1	1477a1	1477.a	1477a	$1$	$1$	\( 7 \cdot 211 \)	\( - 7^{2} \cdot 211 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.210823479$	$1$		$18$	$104$	$-0.522107$	$-24137569/10339$	$[1, 0, 0, -6, 7]$	\(y^2+xy=x^3-6x+7\)
1480.a1	1480a1	1480.a	1480a	$1$	$1$	\( 2^{3} \cdot 5 \cdot 37 \)	\( 2^{8} \cdot 5^{2} \cdot 37 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.081320491$	$1$		$28$	$448$	$-0.231636$	$9483264/925$	$[0, 0, 0, -28, 52]$	\(y^2=x^3-28x+52\)
1483.a1	1483a1	1483.a	1483a	$1$	$1$	\( 1483 \)	\( -1483 \)	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.395288186$	$1$		$14$	$120$	$-0.708076$	$512000/1483$	$[0, 1, 1, 2, 2]$	\(y^2+y=x^3+x^2+2x+2\)

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