Elliptic curves over \(\Q(\sqrt{-7}) \) of conductor 36992.7

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 50000 over imaginary quadratic fields with absolute discriminant 7

Note: The completeness Only modular elliptic curves are included

Results (14 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
36992.7-a1	36992.7-a	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{16} \cdot 17^{2} \)	$3.27880$	$(a), (-a+1), (17)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$0.246793023$	$3.466283336$	5.173302829	\( \frac{6747}{272} a - \frac{4697}{272} \)	\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -1\) , \( -3\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}-{x}-3$
36992.7-a2	36992.7-a	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{17} \cdot 17^{4} \)	$3.27880$	$(a), (-a+1), (17)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$0.246793023$	$1.733141668$	5.173302829	\( -\frac{9868081}{1156} a + \frac{1558107}{68} \)	\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -20 a + 19\) , \( 20 a - 55\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-20a+19\right){x}+20a-55$
36992.7-b1	36992.7-b	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{16} \cdot 17^{2} \)	$3.27880$	$(a), (-a+1), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$0.475768456$	$2.822727762$	4.060743958	\( -\frac{52245}{4} a - \frac{1888217}{68} \)	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -5 a - 9\) , \( 3 a + 15\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a-9\right){x}+3a+15$
36992.7-b2	36992.7-b	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{11} \cdot 17^{4} \)	$3.27880$	$(a), (-a+1), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$0.951536913$	$2.822727762$	4.060743958	\( \frac{23759}{34} a + \frac{63771}{578} \)	\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -a - 5\) , \( 4\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-5\right){x}+4$
36992.7-c1	36992.7-c	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{20} \cdot 17^{4} \)	$3.27880$	$(a), (-a+1), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1.217517423$	$1.213804239$	4.468531274	\( -\frac{60570003}{2312} a - \frac{494720207}{2312} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -39 a - 47\) , \( -257 a - 21\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-39a-47\right){x}-257a-21$
36992.7-c2	36992.7-c	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{22} \cdot 17^{2} \)	$3.27880$	$(a), (-a+1), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$0.608758711$	$2.427608478$	4.468531274	\( \frac{75027}{1088} a + \frac{600895}{1088} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( a - 7\) , \( -9 a + 3\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a-7\right){x}-9a+3$
36992.7-d1	36992.7-d	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{30} \cdot 17^{2} \)	$3.27880$	$(a), (-a+1), (17)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{3} \)	$1$	$1.481513058$	2.239837209	\( \frac{3048625}{1088} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 15 a - 21\) , \( -17 a + 11\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15a-21\right){x}-17a+11$
36992.7-d2	36992.7-d	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{20} \cdot 17^{12} \)	$3.27880$	$(a), (-a+1), (17)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$4$	\( 2^{2} \cdot 3 \)	$1$	$0.246918843$	2.239837209	\( \frac{159661140625}{48275138} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 565 a - 791\) , \( 5593 a - 3619\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(565a-791\right){x}+5593a-3619$
36992.7-d3	36992.7-d	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{24} \cdot 17^{4} \)	$3.27880$	$(a), (-a+1), (17)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$4$	\( 2^{2} \)	$1$	$0.740756529$	2.239837209	\( \frac{8805624625}{2312} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 215 a - 301\) , \( -1785 a + 1155\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(215a-301\right){x}-1785a+1155$
36992.7-d4	36992.7-d	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{22} \cdot 17^{6} \)	$3.27880$	$(a), (-a+1), (17)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{3} \cdot 3 \)	$1$	$0.493837686$	2.239837209	\( \frac{120920208625}{19652} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 515 a - 721\) , \( 6987 a - 4521\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(515a-721\right){x}+6987a-4521$
36992.7-e1	36992.7-e	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{12} \cdot 17^{4} \)	$3.27880$	$(a), (-a+1), (17)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \cdot 3 \)	$0.467741855$	$2.140049118$	9.080132736	\( -\frac{704618595}{18496} a - \frac{339029199}{18496} \)	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -6 a - 18\) , \( -21 a - 31\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-6a-18\right){x}-21a-31$
36992.7-e2	36992.7-e	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{24} \cdot 17^{2} \)	$3.27880$	$(a), (-a+1), (17)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \cdot 3 \)	$0.116935463$	$2.140049118$	9.080132736	\( \frac{38545875}{69632} a + \frac{46531071}{69632} \)	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -6 a + 9\) , \( 6 a + 5\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-6a+9\right){x}+6a+5$
36992.7-f1	36992.7-f	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{14} \cdot 17^{2} \)	$3.27880$	$(a), (-a+1), (17)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$3.518430175$	5.319366428	\( \frac{254667}{68} a - \frac{470825}{68} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -2 a + 6\) , \( -5 a + 3\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+6\right){x}-5a+3$
36992.7-f2	36992.7-f	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	36992.7	\( 2^{7} \cdot 17^{2} \)	\( 2^{7} \cdot 17^{4} \)	$3.27880$	$(a), (-a+1), (17)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2 \)	$1$	$3.518430175$	5.319366428	\( -\frac{48727}{578} a - \frac{598931}{578} \)	\( \bigl[a + 1\) , \( a\) , \( 0\) , \( a + 1\) , \( -a + 1\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(a+1\right){x}-a+1$

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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.