Elliptic curves over \(\Q(\sqrt{-1}) \) of conductor 37570.12

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 100000 over imaginary quadratic fields with absolute discriminant 4

Note: The completeness Only modular elliptic curves are included

Results (10 matches)

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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
37570.12-a1	37570.12-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2 \cdot 5^{4} \cdot 13^{4} \cdot 17^{8} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$1$	$0.284528231$	1.138112925	\( -\frac{1460607388319513}{10317661250} a - \frac{23604896084583241}{10317661250} \)	\( \bigl[i\) , \( i - 1\) , \( i\) , \( -2134 i - 443\) , \( -34034 i + 18902\bigr] \)	${y}^2+i{x}{y}+i{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-2134i-443\right){x}-34034i+18902$
37570.12-a2	37570.12-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2 \cdot 5 \cdot 13 \cdot 17^{14} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2^{2} \)	$1$	$0.284528231$	1.138112925	\( -\frac{20757052235979591}{906848467330} a - \frac{16117571297539447}{906848467330} \)	\( \bigl[i\) , \( i - 1\) , \( i\) , \( -524 i - 933\) , \( 9198 i + 10670\bigr] \)	${y}^2+i{x}{y}+i{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-524i-933\right){x}+9198i+10670$
37570.12-a3	37570.12-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \cdot 17^{10} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{5} \)	$1$	$0.569056462$	1.138112925	\( \frac{977034984177}{705752450} a + \frac{241328100032}{352876225} \)	\( \bigl[i\) , \( i - 1\) , \( i\) , \( -129 i - 48\) , \( -422 i + 530\bigr] \)	${y}^2+i{x}{y}+i{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-129i-48\right){x}-422i+530$
37570.12-a4	37570.12-a	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2^{4} \cdot 5 \cdot 13 \cdot 17^{8} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$1$	$1.138112925$	1.138112925	\( -\frac{37220008}{18785} a + \frac{85353791}{75140} \)	\( \bigl[i\) , \( i - 1\) , \( i\) , \( 21 i + 32\) , \( -74 i + 22\bigr] \)	${y}^2+i{x}{y}+i{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(21i+32\right){x}-74i+22$
37570.12-b1	37570.12-b	$6$	$18$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2^{18} \cdot 5^{9} \cdot 13 \cdot 17^{6} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$0.233010375$	2.097093378	\( -\frac{276861163011391}{13000000000} a - \frac{33515586556057}{812500000} \)	\( \bigl[1\) , \( -i\) , \( i\) , \( 939 i + 1471\) , \( 18545 i - 20896\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-i{x}^{2}+\left(939i+1471\right){x}+18545i-20896$
37570.12-b2	37570.12-b	$6$	$18$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2^{6} \cdot 5^{3} \cdot 13^{3} \cdot 17^{6} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3Cs	$1$	\( 2^{2} \cdot 3 \)	$1$	$0.699031126$	2.097093378	\( \frac{37525044319}{2197000} a - \frac{7169596274}{274625} \)	\( \bigl[1\) , \( -i\) , \( i\) , \( 179 i + 6\) , \( 658 i + 757\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-i{x}^{2}+\left(179i+6\right){x}+658i+757$
37570.12-b3	37570.12-b	$6$	$18$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2^{3} \cdot 5^{6} \cdot 13^{6} \cdot 17^{6} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3Cs	$1$	\( 2^{3} \cdot 3 \)	$1$	$0.349515563$	2.097093378	\( \frac{133816114442969}{301675562500} a - \frac{19082395919017}{301675562500} \)	\( \bigl[1\) , \( -i\) , \( i\) , \( 109 i - 224\) , \( -194 i + 3077\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-i{x}^{2}+\left(109i-224\right){x}-194i+3077$
37570.12-b4	37570.12-b	$6$	$18$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2^{9} \cdot 5^{18} \cdot 13^{2} \cdot 17^{6} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{3} \cdot 3^{2} \)	$1$	$0.116505187$	2.097093378	\( -\frac{8418015312387897223}{20629882812500000} a + \frac{2783266907131437289}{20629882812500000} \)	\( \bigl[1\) , \( -i\) , \( i\) , \( -901 i + 2031\) , \( -10607 i - 78368\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-i{x}^{2}+\left(-901i+2031\right){x}-10607i-78368$
37570.12-b5	37570.12-b	$6$	$18$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2^{2} \cdot 5 \cdot 13 \cdot 17^{6} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$1$	$2.097093378$	2.097093378	\( \frac{31409}{130} a + \frac{101344}{65} \)	\( \bigl[1\) , \( -i\) , \( i\) , \( -11 i + 1\) , \( i + 2\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-i{x}^{2}+\left(-11i+1\right){x}+i+2$
37570.12-b6	37570.12-b	$6$	$18$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	37570.12	\( 2 \cdot 5 \cdot 13 \cdot 17^{2} \)	\( 2 \cdot 5^{2} \cdot 13^{2} \cdot 17^{6} \)	$2.48816$	$(a+1), (2a+1), (2a+3), (a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{3} \)	$1$	$1.048546689$	2.097093378	\( \frac{4406742137}{8450} a + \frac{1310300809}{8450} \)	\( \bigl[1\) , \( -i\) , \( i\) , \( -126 i + 36\) , \( -225 i + 498\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-i{x}^{2}+\left(-126i+36\right){x}-225i+498$

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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.