Elliptic curves over Q(−3)\Q(\sqrt{-3}) of conductor 61731.3

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Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
61731.3-a1	61731.3-a	$5$	$9$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{6} \cdot 19^{8}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B[2]	$1$	$2^{3}$	$1.182529221$	$0.123884704$	2.706567867	$-\frac{50357871050752}{19}$	$\bigl[0$ , $-a - 1$ , $1$ , $36929 a + 11540$ , $1351376 a - 4007862\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(36929a+11540\right){x}+1351376a-4007862$
61731.3-a2	61731.3-a	$5$	$9$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{6} \cdot 19^{16}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B[2]	$1$	$2^{3}$	$1.182529221$	$0.123884704$	2.706567867	$\frac{14306161739497472}{322687697779} a - \frac{27123251845038080}{322687697779}$	$\bigl[0$ , $-a - 1$ , $1$ , $7731 a - 4308$ , $-195516 a - 34450\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(7731a-4308\right){x}-195516a-34450$
61731.3-a3	61731.3-a	$5$	$9$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{6} \cdot 19^{16}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B[2]	$1$	$2^{3}$	$1.182529221$	$0.123884704$	2.706567867	$-\frac{14306161739497472}{322687697779} a - \frac{12817090105540608}{322687697779}$	$\bigl[0$ , $-a - 1$ , $1$ , $-6699 a - 18$ , $251154 a - 123514\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6699a-18\right){x}+251154a-123514$
61731.3-a4	61731.3-a	$5$	$9$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{6} \cdot 19^{12}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3Cs[2]	$1$	$2^{3}$	$0.394176407$	$0.371654113$	2.706567867	$-\frac{89915392}{6859}$	$\bigl[0$ , $-a - 1$ , $1$ , $141 a - 588$ , $2064 a - 5467\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(141a-588\right){x}+2064a-5467$
61731.3-a5	61731.3-a	$5$	$9$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{6} \cdot 19^{8}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B[2]	$1$	$2^{3}$	$0.131392135$	$1.114962340$	2.706567867	$\frac{32768}{19}$	$\bigl[0$ , $-a - 1$ , $1$ , $43 a - 32$ , $-16 a + 20\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(43a-32\right){x}-16a+20$
61731.3-b1	61731.3-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{7} \cdot 19^{16}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{5}$	$1$	$0.108104655$	0.998628029	$\frac{7240152655469734}{50950689123} a - \frac{5512832666599067}{50950689123}$	$\bigl[a + 1$ , $-a - 1$ , $a$ , $-9143 a + 10240$ , $68164 a + 329450\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-9143a+10240\right){x}+68164a+329450$
61731.3-b2	61731.3-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{8} \cdot 19^{14}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{6}$	$1$	$0.216209310$	0.998628029	$\frac{67419143}{390963}$	$\bigl[a + 1$ , $-a - 1$ , $a$ , $-128 a + 535$ , $-4787 a + 13769\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-128a+535\right){x}-4787a+13769$
61731.3-b3	61731.3-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{8} \cdot 19^{8}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$1$	$0.864837242$	0.998628029	$\frac{389017}{57}$	$\bigl[a + 1$ , $-a - 1$ , $a$ , $22 a - 95$ , $103 a - 271\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(22a-95\right){x}+103a-271$
61731.3-b4	61731.3-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{10} \cdot 19^{10}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$1$	$0.432418621$	0.998628029	$\frac{30664297}{3249}$	$\bigl[a + 1$ , $-a - 1$ , $a$ , $97 a - 410$ , $-980 a + 2978\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(97a-410\right){x}-980a+2978$
61731.3-b5	61731.3-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{7} \cdot 19^{16}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{5}$	$1$	$0.108104655$	0.998628029	$-\frac{7240152655469734}{50950689123} a + \frac{1727319988870667}{50950689123}$	$\bigl[a + 1$ , $-a - 1$ , $a$ , $5287 a + 5950$ , $-322106 a + 404732\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5287a+5950\right){x}-322106a+404732$
61731.3-b6	61731.3-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{14} \cdot 19^{8}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{4}$	$1$	$0.216209310$	0.998628029	$\frac{115714886617}{1539}$	$\bigl[a + 1$ , $-a - 1$ , $a$ , $1522 a - 6395$ , $-66245 a + 194783\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1522a-6395\right){x}-66245a+194783$
61731.3-c1	61731.3-c	$4$	$6$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{3} \cdot 19^{9}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B[2]	$1$	$2^{2}$	$1$	$1.078091533$	1.244872874	$\frac{29840721}{6859} a - \frac{35267232}{6859}$	$\bigl[a + 1$ , $-a - 1$ , $0$ , $-16 a - 40$ , $-81 a - 75\bigr]$	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-16a-40\right){x}-81a-75$
61731.3-c2	61731.3-c	$4$	$6$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{3} \cdot 19^{12}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B[2]	$1$	$2^{3}$	$1$	$0.539045766$	1.244872874	$-\frac{36038181633}{47045881} a - \frac{39546962313}{47045881}$	$\bigl[a + 1$ , $-a - 1$ , $0$ , $-146 a + 145$ , $222 a - 1131\bigr]$	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-146a+145\right){x}+222a-1131$
61731.3-c3	61731.3-c	$4$	$6$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{9} \cdot 19^{7}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B[2]	$1$	$2^{2}$	$1$	$1.078091533$	1.244872874	$-\frac{9153}{19} a + \frac{36801}{19}$	$\bigl[a + 1$ , $0$ , $a + 1$ , $18 a + 27$ , $-45 a + 20\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(18a+27\right){x}-45a+20$
61731.3-c4	61731.3-c	$4$	$6$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{9} \cdot 19^{8}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B[2]	$1$	$2^{3}$	$1$	$0.539045766$	1.244872874	$-\frac{363527109}{361} a + \frac{287391186}{361}$	$\bigl[a + 1$ , $0$ , $a + 1$ , $183 a + 417$ , $-4773 a + 4499\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(183a+417\right){x}-4773a+4499$
61731.3-d1	61731.3-d	$2$	$5$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{10} \cdot 19^{16}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.2	$1$	$2^{2} \cdot 5$	$1$	$0.035996263$	0.831298097	$-\frac{9358714467168256}{22284891}$	$\bigl[0$ , $-a - 1$ , $1$ , $65856 a - 276591$ , $18858975 a - 54638371\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(65856a-276591\right){x}+18858975a-54638371$
61731.3-d2	61731.3-d	$2$	$5$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{26} \cdot 19^{8}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.1	$1$	$2^{2}$	$1$	$0.179981317$	0.831298097	$\frac{841232384}{1121931}$	$\bigl[0$ , $-a - 1$ , $1$ , $-294 a + 1239$ , $6225 a - 19261\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-294a+1239\right){x}+6225a-19261$
61731.3-e1	61731.3-e	$1$	$1$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{12} \cdot 19^{10}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2^{2}$	$1$	$0.371111235$	1.714089374	$-\frac{1024000}{513} a - \frac{512000}{171}$	$\bigl[0$ , $-a - 1$ , $1$ , $366 a - 450$ , $-4386 a + 3212\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(366a-450\right){x}-4386a+3212$
61731.3-f1	61731.3-f	$1$	$1$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{10} \cdot 19^{8}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓						$1$	$2^{2}$	$1$	$0.705796155$	3.259932801	$-\frac{1404928}{171}$	$\bigl[0$ , $a + 1$ , $1$ , $36 a - 147$ , $-279 a + 677\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(36a-147\right){x}-279a+677$
61731.3-g1	61731.3-g	$2$	$2$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{3} \cdot 19^{13}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$2.769040107$	$0.356166928$	4.555249792	$\frac{5845799184}{130321} a - \frac{4133064549}{130321}$	$\bigl[1$ , $-a + 1$ , $1$ , $238 a - 817$ , $3795 a - 8972\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(238a-817\right){x}+3795a-8972$
61731.3-g2	61731.3-g	$2$	$2$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{3} \cdot 19^{11}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$1.384520053$	$0.712333856$	4.555249792	$\frac{61536}{361} a - \frac{40335}{361}$	$\bigl[1$ , $-a + 1$ , $1$ , $-42 a - 2$ , $343 a - 394\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-42a-2\right){x}+343a-394$
61731.3-h1	61731.3-h	$2$	$2$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{9} \cdot 19^{7}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$0.896333780$	4.139988395	$\frac{5845799184}{130321} a - \frac{4133064549}{130321}$	$\bigl[a$ , $-a + 1$ , $a$ , $7 a - 118$ , $-7 a + 466\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7a-118\right){x}-7a+466$
61731.3-h2	61731.3-h	$2$	$2$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	61731.3	$3^{2} \cdot 19^{3}$	$3^{9} \cdot 19^{5}$	$2.43964$	$(-2a+1), (-5a+3), (-5a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$1$	$1.792667560$	4.139988395	$\frac{61536}{361} a - \frac{40335}{361}$	$\bigl[a$ , $-a + 1$ , $a$ , $-8 a + 2$ , $-16 a + 25\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a+2\right){x}-16a+25$

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  *The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.