Elliptic curves over \(\Q(\sqrt{37}) \) of conductor 441.9

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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
441.9-a1	441.9-a	$2$	$2$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{10} \cdot 7^{3} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$9.194557460$	3.023151870	\( -\frac{247504499}{81} a - \frac{67864571}{9} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -79 a - 205\) , \( 695 a + 1764\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-79a-205\right){x}+695a+1764$
441.9-a2	441.9-a	$2$	$2$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{8} \cdot 7^{3} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2^{2} \)	$1$	$4.597278730$	3.023151870	\( \frac{498881612022497}{9} a + 140872042480094 \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -1274 a - 3240\) , \( 43503 a + 110552\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-1274a-3240\right){x}+43503a+110552$
441.9-b1	441.9-b	$2$	$5$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( 3^{6} \cdot 7^{6} \)	$2.49086$	$(a+2), (a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3, 5$	2Cn, 3Ns, 5B.4.1	$1$	\( 1 \)	$1$	$14.82684924$	2.437519000	\( 4096 \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 48 a - 168\) , \( -186 a + 658\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(48a-168\right){x}-186a+658$
441.9-b2	441.9-b	$2$	$5$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( 3^{6} \cdot 7^{6} \)	$2.49086$	$(a+2), (a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3, 5$	2Cn, 3Ns, 5B.4.2	$25$	\( 1 \)	$1$	$0.593073969$	2.437519000	\( 38477541376 \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 10198 a - 36148\) , \( 990538 a - 3507924\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(10198a-36148\right){x}+990538a-3507924$
441.9-c1	441.9-c	$2$	$2$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{20} \cdot 7^{7} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$3.948996005$	1.298421888	\( -\frac{1449791608112}{33480783} a - \frac{325818657413}{3720087} \)	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 1174 a - 4175\) , \( -36966 a + 130912\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(1174a-4175\right){x}-36966a+130912$
441.9-c2	441.9-c	$2$	$2$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( 3^{13} \cdot 7^{8} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$3.948996005$	1.298421888	\( \frac{1020689116688854}{107163} a + \frac{293830214716873}{11907} \)	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 18579 a - 66040\) , \( -2414069 a + 8550255\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(18579a-66040\right){x}-2414069a+8550255$
441.9-d1	441.9-d	$4$	$6$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{8} \cdot 7^{9} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{3} \)	$1$	$3.763493638$	1.237429085	\( -\frac{48155111}{3087} a + \frac{18937350}{343} \)	\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 80 a - 277\) , \( -694 a + 2458\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(80a-277\right){x}-694a+2458$
441.9-d2	441.9-d	$4$	$6$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{12} \cdot 7^{7} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{3} \)	$1$	$3.763493638$	1.237429085	\( -\frac{26608457}{5103} a + \frac{10488103}{567} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 33 a + 83\) , \( 1128 a + 2866\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(33a+83\right){x}+1128a+2866$
441.9-d3	441.9-d	$4$	$6$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{18} \cdot 7^{8} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{4} \)	$1$	$1.881746819$	1.237429085	\( \frac{82104162493}{26040609} a + \frac{23195606656}{2893401} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -692 a - 1757\) , \( 14400 a + 36592\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-692a-1757\right){x}+14400a+36592$
441.9-d4	441.9-d	$4$	$6$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{10} \cdot 7^{12} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{4} \)	$1$	$1.881746819$	1.237429085	\( \frac{271552460437}{9529569} a + \frac{76675641283}{1058841} \)	\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 15 a - 77\) , \( -1666 a + 5788\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(15a-77\right){x}-1666a+5788$
441.9-e1	441.9-e	$4$	$4$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{8} \cdot 7^{7} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$4.609070673$	1.515453102	\( -\frac{33776}{63} a + \frac{17839}{7} \)	\( \bigl[a\) , \( -a\) , \( 1\) , \( 51 a - 182\) , \( 341 a - 1208\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(51a-182\right){x}+341a-1208$
441.9-e2	441.9-e	$4$	$4$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( 3^{10} \cdot 7^{8} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$4$	\( 2^{4} \)	$1$	$2.304535336$	1.515453102	\( -\frac{6273137672}{3969} a + \frac{2470185541}{441} \)	\( \bigl[a\) , \( -a\) , \( 1\) , \( 776 a - 2752\) , \( 21667 a - 76732\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(776a-2752\right){x}+21667a-76732$
441.9-e3	441.9-e	$4$	$4$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( 3^{8} \cdot 7^{7} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2^{3} \)	$1$	$1.152267668$	1.515453102	\( -\frac{957961400352382}{63} a + \frac{376945173009773}{7} \)	\( \bigl[a\) , \( -a\) , \( 1\) , \( 12441 a - 44072\) , \( 1335073 a - 4727986\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(12441a-44072\right){x}+1335073a-4727986$
441.9-e4	441.9-e	$4$	$4$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{14} \cdot 7^{10} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2^{4} \)	$1$	$0.576133834$	1.515453102	\( \frac{27348970029058}{15752961} a + \frac{7718451347197}{1750329} \)	\( \bigl[a\) , \( -a\) , \( 1\) , \( 711 a - 2552\) , \( 24885 a - 88194\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(711a-2552\right){x}+24885a-88194$
441.9-f1	441.9-f	$2$	$2$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{10} \cdot 7^{9} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$0.652746375$	0.214621686	\( -\frac{247504499}{81} a - \frac{67864571}{9} \)	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -176 a - 530\) , \( -2689 a - 7103\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-176a-530\right){x}-2689a-7103$
441.9-f2	441.9-f	$2$	$2$	\(\Q(\sqrt{37}) \)	$2$	$[2, 0]$	441.9	\( 3^{2} \cdot 7^{2} \)	\( - 3^{8} \cdot 7^{9} \)	$2.49086$	$(a+2), (a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2^{2} \)	$1$	$0.326373187$	0.214621686	\( \frac{498881612022497}{9} a + 140872042480094 \)	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -3021 a - 7720\) , \( -162246 a - 412881\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-3021a-7720\right){x}-162246a-412881$

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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.