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
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	$1$	$1$		$0$	$5$	$0.496709$	$-52893159101157376/11$	$[0, -1, 1, -7820, -263580]$	\(y^2+y=x^3-x^2-7820x-263580\)
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	$1$	$1$		$0$	$3$	$0.033439$	$-50357871050752/19$	$[0, 1, 1, -769, -8470]$	\(y^2+y=x^3+x^2-769x-8470\)
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.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	$1$	$1$		$0$	$14$	$0.289794$	$-1064019559329/125497034$	$[1, -1, 1, -213, -1257]$	\(y^2+xy+y=x^3-x^2-213x-1257\)
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	$1$	$1$		$0$	$3$	$0.052148$	$-12288000$	$[0, 0, 1, -270, -1708]$	\(y^2+y=x^3-270x-1708\)
35.a1	35a2	35.a	35a	$3$	$9$	\( 5 \cdot 7 \)	\( - 5^{9} \cdot 7 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	9.24.0.3	3B.1.2	$1$	$1$		$0$	$6$	$0.127462$	$-250523582464/13671875$	$[0, 1, 1, -131, -650]$	\(y^2+y=x^3+x^2-131x-650\)
37.a1	37a1	37.a	37a	$1$	$1$	\( 37 \)	\( 37 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.051111408$	$1$		$10$	$2$	$-0.996542$	$110592/37$	$[0, 0, 1, -1, 0]$	\(y^2+y=x^3-x\)
37.b1	37b2	37.b	37b	$3$	$9$	\( 37 \)	\( 37 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	27.72.0.2	3B.1.2	$1$	$1$		$0$	$6$	$0.222082$	$727057727488000/37$	$[0, 1, 1, -1873, -31833]$	\(y^2+y=x^3+x^2-1873x-31833\)
38.a1	38a2	38.a	38a	$3$	$9$	\( 2 \cdot 19 \)	\( - 2^{27} \cdot 19 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	27.72.0.2	3B.1.2	$1$	$1$		$0$	$18$	$0.485169$	$-69173457625/2550136832$	$[1, 0, 1, -86, -2456]$	\(y^2+xy+y=x^3-86x-2456\)
38.b1	38b2	38.b	38b	$2$	$5$	\( 2 \cdot 19 \)	\( - 2 \cdot 19^{5} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$5$	5.24.0.3	5B.1.2	$1$	$1$		$0$	$10$	$0.017785$	$-37966934881/4952198$	$[1, 1, 1, -70, -279]$	\(y^2+xy+y=x^3+x^2-70x-279\)
43.a1	43a1	43.a	43a	$1$	$1$	\( 43 \)	\( -43 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.062816507$	$1$		$10$	$2$	$-1.004431$	$-4096/43$	$[0, 1, 1, 0, 0]$	\(y^2+y=x^3+x^2\)
44.a1	44a2	44.a	44a	$2$	$3$	\( 2^{2} \cdot 11 \)	\( - 2^{8} \cdot 11^{3} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3$	3.8.0.2	3B.1.2	$1$	$1$		$0$	$6$	$-0.102988$	$-199794688/1331$	$[0, 1, 0, -77, -289]$	\(y^2=x^3+x^2-77x-289\)
50.a1	50a2	50.a	50a	$4$	$15$	\( 2 \cdot 5^{2} \)	\( - 2^{3} \cdot 5^{4} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 3, 5$	8.2.0.1, 3.8.0.2, 5.24.0.4	3B.1.2, 5B.1.3	$1$	$1$		$0$	$6$	$-0.176793$	$-349938025/8$	$[1, 0, 1, -126, -552]$	\(y^2+xy+y=x^3-126x-552\)
50.a4	50a4	50.a	50a	$4$	$15$	\( 2 \cdot 5^{2} \)	\( - 2^{15} \cdot 5^{8} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 3, 5$	8.2.0.1, 3.8.0.2, 5.24.0.2	3B.1.2, 5B.1.4	$1$	$1$		$0$	$30$	$0.627926$	$46969655/32768$	$[1, 0, 1, 549, -2202]$	\(y^2+xy+y=x^3+549x-2202\)
50.b1	50b4	50.b	50b	$4$	$15$	\( 2 \cdot 5^{2} \)	\( - 2^{3} \cdot 5^{10} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 3, 5$	8.2.0.1, 3.4.0.1, 5.24.0.3	3B, 5B.1.2	$1$	$1$		$0$	$30$	$0.627926$	$-349938025/8$	$[1, 1, 1, -3138, -68969]$	\(y^2+xy+y=x^3+x^2-3138x-68969\)
50.b2	50b3	50.b	50b	$4$	$15$	\( 2 \cdot 5^{2} \)	\( - 2 \cdot 5^{10} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 3, 5$	8.2.0.1, 3.4.0.1, 5.24.0.3	3B, 5B.1.2	$1$	$1$		$0$	$10$	$0.078619$	$-25/2$	$[1, 1, 1, -13, -219]$	\(y^2+xy+y=x^3+x^2-13x-219\)
51.a1	51a2	51.a	51a	$2$	$3$	\( 3 \cdot 17 \)	\( - 3 \cdot 17^{3} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	3.8.0.2	3B.1.2	$1$	$1$		$0$	$6$	$-0.259223$	$-23100424192/14739$	$[0, 1, 1, -59, -196]$	\(y^2+y=x^3+x^2-59x-196\)
53.a1	53a1	53.a	53a	$1$	$1$	\( 53 \)	\( -53 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.092981484$	$1$		$8$	$2$	$-0.988547$	$3375/53$	$[1, -1, 1, 0, 0]$	\(y^2+xy+y=x^3-x^2\)
54.a1	54a2	54.a	54a	$3$	$9$	\( 2 \cdot 3^{3} \)	\( - 2^{9} \cdot 3^{11} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3$	9.72.0.11	3B.1.2	$1$	$1$		$0$	$18$	$0.234102$	$-1167051/512$	$[1, -1, 0, -123, -667]$	\(y^2+xy=x^3-x^2-123x-667\)
54.b1	54b2	54.b	54b	$3$	$9$	\( 2 \cdot 3^{3} \)	\( - 2 \cdot 3^{9} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3$	9.72.0.12	3B.1.2	$1$	$1$		$0$	$6$	$-0.315205$	$-132651/2$	$[1, -1, 1, -29, -53]$	\(y^2+xy+y=x^3-x^2-29x-53\)
57.a1	57a1	57.a	57a	$1$	$1$	\( 3 \cdot 19 \)	\( - 3^{2} \cdot 19 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.037574592$	$1$		$12$	$4$	$-0.836549$	$-1404928/171$	$[0, -1, 1, -2, 2]$	\(y^2+y=x^3-x^2-2x+2\)
57.b1	57c2	57.b	57c	$2$	$5$	\( 3 \cdot 19 \)	\( - 3^{2} \cdot 19^{5} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$5$	5.24.0.3	5B.1.2	$1$	$1$		$0$	$60$	$0.651407$	$-9358714467168256/22284891$	$[0, 1, 1, -4390, -113432]$	\(y^2+y=x^3+x^2-4390x-113432\)
58.a1	58a1	58.a	58a	$1$	$1$	\( 2 \cdot 29 \)	\( - 2^{2} \cdot 29 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.042420307$	$1$		$12$	$4$	$-0.902228$	$-185193/116$	$[1, -1, 0, -1, 1]$	\(y^2+xy=x^3-x^2-x+1\)
58.b1	58b2	58.b	58b	$2$	$5$	\( 2 \cdot 29 \)	\( - 2^{2} \cdot 29^{5} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$5$	5.24.0.3	5B.1.2	$1$	$1$		$0$	$20$	$0.347164$	$-10418796526321/82044596$	$[1, 1, 1, -455, -3951]$	\(y^2+xy+y=x^3+x^2-455x-3951\)
61.a1	61a1	61.a	61a	$1$	$1$	\( 61 \)	\( -61 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.079187731$	$1$		$8$	$2$	$-0.905458$	$-912673/61$	$[1, 0, 0, -2, 1]$	\(y^2+xy=x^3-2x+1\)
67.a1	67a1	67.a	67a	$1$	$1$	\( 67 \)	\( -67 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$1$	$1$		$0$	$5$	$-0.675266$	$-207474688/67$	$[0, 1, 1, -12, -21]$	\(y^2+y=x^3+x^2-12x-21\)
75.a1	75c2	75.a	75c	$2$	$5$	\( 3 \cdot 5^{2} \)	\( - 3 \cdot 5^{10} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	5.24.0.3	5B.1.2	$1$	$1$		$0$	$30$	$0.211621$	$-102400/3$	$[0, 1, 1, -208, -1256]$	\(y^2+y=x^3+x^2-208x-1256\)
75.c1	75a1	75.c	75a	$2$	$5$	\( 3 \cdot 5^{2} \)	\( - 3 \cdot 5^{4} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	5.24.0.4	5B.1.3	$1$	$1$		$0$	$6$	$-0.593099$	$-102400/3$	$[0, -1, 1, -8, -7]$	\(y^2+y=x^3-x^2-8x-7\)
75.c2	75a2	75.c	75a	$2$	$5$	\( 3 \cdot 5^{2} \)	\( - 3^{5} \cdot 5^{8} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	5.24.0.2	5B.1.4	$1$	$1$		$0$	$30$	$0.211621$	$20480/243$	$[0, -1, 1, 42, 443]$	\(y^2+y=x^3-x^2+42x+443\)
76.a1	76a1	76.a	76a	$1$	$1$	\( 2^{2} \cdot 19 \)	\( - 2^{8} \cdot 19 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$1$	$1$		$0$	$6$	$-0.440931$	$-4194304/19$	$[0, -1, 0, -21, -31]$	\(y^2=x^3-x^2-21x-31\)
77.a1	77a1	77.a	77a	$1$	$1$	\( 7 \cdot 11 \)	\( - 7^{2} \cdot 11 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.098027979$	$1$		$8$	$4$	$-0.786654$	$884736/539$	$[0, 0, 1, 2, 0]$	\(y^2+y=x^3+2x\)
77.b3	77b2	77.b	77b	$3$	$9$	\( 7 \cdot 11 \)	\( - 7^{2} \cdot 11^{9} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	9.24.0.3	3B.1.2	$1$	$1$		$0$	$60$	$0.803613$	$9463555063808/115539436859$	$[0, 1, 1, 441, -15815]$	\(y^2+y=x^3+x^2+441x-15815\)
79.a1	79a1	79.a	79a	$1$	$1$	\( 79 \)	\( 79 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.097664210$	$1$		$8$	$2$	$-0.895041$	$912673/79$	$[1, 1, 1, -2, 0]$	\(y^2+xy+y=x^3+x^2-2x\)
83.a1	83a1	83.a	83a	$1$	$1$	\( 83 \)	\( -83 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.177292294$	$1$		$6$	$2$	$-0.943868$	$103823/83$	$[1, 1, 1, 1, 0]$	\(y^2+xy+y=x^3+x^2+x\)
88.a1	88a1	88.a	88a	$1$	$1$	\( 2^{3} \cdot 11 \)	\( - 2^{8} \cdot 11 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.040264364$	$1$		$14$	$8$	$-0.629211$	$-27648/11$	$[0, 0, 0, -4, 4]$	\(y^2=x^3-4x+4\)
89.a1	89a1	89.a	89a	$1$	$1$	\( 89 \)	\( -89 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.112104881$	$1$		$6$	$2$	$-0.926876$	$-117649/89$	$[1, 1, 1, -1, 0]$	\(y^2+xy+y=x^3+x^2-x\)
91.a1	91a1	91.a	91a	$1$	$1$	\( 7 \cdot 13 \)	\( - 7 \cdot 13 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.142392150$	$1$		$6$	$4$	$-0.936330$	$110592/91$	$[0, 0, 1, 1, 0]$	\(y^2+y=x^3+x\)
91.b1	91b3	91.b	91b	$3$	$9$	\( 7 \cdot 13 \)	\( - 7^{9} \cdot 13 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	9.24.0.3	3B.1.2	$0.117693898$	$1$		$10$	$36$	$0.360104$	$-178643795968/524596891$	$[0, 1, 1, -117, -1245]$	\(y^2+y=x^3+x^2-117x-1245\)
92.a1	92b1	92.a	92b	$1$	$1$	\( 2^{2} \cdot 23 \)	\( - 2^{4} \cdot 23 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.049808397$	$1$		$12$	$6$	$-0.821413$	$-6912/23$	$[0, 0, 0, -1, 1]$	\(y^2=x^3-x+1\)
92.b1	92a2	92.b	92a	$2$	$3$	\( 2^{2} \cdot 23 \)	\( - 2^{4} \cdot 23^{3} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3$	3.8.0.2	3B.1.2	$1$	$1$		$0$	$6$	$-0.269963$	$-42592000/12167$	$[0, 1, 0, -18, -43]$	\(y^2=x^3+x^2-18x-43\)
99.d1	99d3	99.d	99d	$3$	$25$	\( 3^{2} \cdot 11 \)	\( - 3^{6} \cdot 11 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	25.60.0.2	5B.4.2	$1$	$1$		$0$	$150$	$1.046015$	$-52893159101157376/11$	$[0, 0, 1, -70383, 7187035]$	\(y^2+y=x^3-70383x+7187035\)
99.d2	99d2	99.d	99d	$3$	$25$	\( 3^{2} \cdot 11 \)	\( - 3^{6} \cdot 11^{5} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	5.60.0.1	5Cs.4.1	$1$	$1$		$0$	$30$	$0.241296$	$-122023936/161051$	$[0, 0, 1, -93, 625]$	\(y^2+y=x^3-93x+625\)
99.d3	99d1	99.d	99d	$3$	$25$	\( 3^{2} \cdot 11 \)	\( - 3^{6} \cdot 11 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	25.60.0.1	5B.4.1	$1$	$1$		$0$	$6$	$-0.563422$	$-4096/11$	$[0, 0, 1, -3, -5]$	\(y^2+y=x^3-3x-5\)
101.a1	101a1	101.a	101a	$1$	$1$	\( 101 \)	\( 101 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.164703452$	$1$		$6$	$2$	$-0.916363$	$262144/101$	$[0, 1, 1, -1, -1]$	\(y^2+y=x^3+x^2-x-1\)
104.a1	104a1	104.a	104a	$1$	$1$	\( 2^{3} \cdot 13 \)	\( - 2^{11} \cdot 13 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$1$	$1$		$0$	$8$	$-0.393032$	$-235298/13$	$[0, 1, 0, -16, -32]$	\(y^2=x^3+x^2-16x-32\)
106.a1	106b1	106.a	106b	$1$	$1$	\( 2 \cdot 53 \)	\( - 2^{4} \cdot 53 \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$0.068912680$	$1$		$10$	$8$	$-0.641726$	$-47045881/848$	$[1, 1, 0, -7, 5]$	\(y^2+xy=x^3+x^2-7x+5\)
106.b1	106d1	106.b	106d	$1$	$1$	\( 2 \cdot 53 \)	\( - 2^{5} \cdot 53 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓					$1$	$1$		$0$	$10$	$-0.446259$	$-2305199161/1696$	$[1, 1, 0, -27, -67]$	\(y^2+xy=x^3+x^2-27x-67\)
106.c1	106a2	106.c	106a	$2$	$3$	\( 2 \cdot 53 \)	\( - 2 \cdot 53^{3} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	3.8.0.2	3B.1.2	$1$	$1$		$0$	$18$	$-0.263907$	$-81182737/297754$	$[1, 0, 0, -9, -29]$	\(y^2+xy=x^3-9x-29\)
106.d1	106c2	106.d	106c	$2$	$3$	\( 2 \cdot 53 \)	\( - 2^{8} \cdot 53^{3} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	3.8.0.2	3B.1.2	$1$	$1$		$0$	$144$	$0.977397$	$-1646982616152408625/38112512$	$[1, 0, 0, -24603, -1487407]$	\(y^2+xy=x^3-24603x-1487407\)
108.a1	108a2	108.a	108a	$2$	$3$	\( 2^{2} \cdot 3^{3} \)	\( - 2^{8} \cdot 3^{9} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$3$	27.648.18.4	3B.1.2	$1$	$1$		$0$	$18$	$-0.035060$	$0$	$[0, 0, 0, 0, -108]$	\(y^2=x^3-108\)

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