Elliptic curves over \(\Q(\sqrt{-3}) \) of conductor 53361.3

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 150000 over imaginary quadratic fields with absolute discriminant 3

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
53361.3-a1	53361.3-a	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{13} \cdot 7^{2} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	\( 2^{2} \)	$0.174978614$	$2.022470152$	3.269086780	\( \frac{210097}{891} a + \frac{1457662}{891} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -3 a + 12\) , \( -2 a + 10\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a+12\right){x}-2a+10$
53361.3-b1	53361.3-b	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{9} \cdot 7^{4} \cdot 11^{6} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1[2]	$1$	\( 2 \cdot 3^{2} \)	$1$	$0.940340520$	2.171623411	\( -\frac{2003475}{1331} a + \frac{1869918}{1331} \)	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -17 a + 56\) , \( 74 a + 88\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-17a+56\right){x}+74a+88$
53361.3-b2	53361.3-b	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{3} \cdot 7^{4} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1[2]	$1$	\( 2 \cdot 3 \)	$1$	$2.821021562$	2.171623411	\( -\frac{1176039}{11} a + \frac{1096065}{11} \)	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -7 a - 9\) , \( 15 a + 6\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7a-9\right){x}+15a+6$
53361.3-c1	53361.3-c	$3$	$25$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 7^{6} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.2	$1$	\( 2 \)	$10.54640577$	$0.080807988$	3.936299486	\( -\frac{52893159101157376}{11} \)	\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 187689 a - 70383\) , \( 16072854 a + 13489487\bigr] \)	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(187689a-70383\right){x}+16072854a+13489487$
53361.3-c2	53361.3-c	$3$	$25$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 7^{6} \cdot 11^{10} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5Cs.4.1	$1$	\( 2 \)	$2.109281154$	$0.404039943$	3.936299486	\( -\frac{122023936}{161051} \)	\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 249 a - 93\) , \( 1524 a + 1067\bigr] \)	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(249a-93\right){x}+1524a+1067$
53361.3-c3	53361.3-c	$3$	$25$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 7^{6} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.1	$1$	\( 2 \)	$0.421856230$	$2.020199715$	3.936299486	\( -\frac{4096}{11} \)	\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 9 a - 3\) , \( -6 a - 13\bigr] \)	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a-3\right){x}-6a-13$
53361.3-d1	53361.3-d	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{3} \cdot 7^{10} \cdot 11^{6} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B[2]	$1$	\( 2 \)	$1$	$0.615597373$	1.421661237	\( -\frac{2003475}{1331} a + \frac{1869918}{1331} \)	\( \bigl[a\) , \( a\) , \( a\) , \( -121 a + 12\) , \( 500 a + 1\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-121a+12\right){x}+500a+1$
53361.3-d2	53361.3-d	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{9} \cdot 7^{10} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B[2]	$1$	\( 2 \)	$1$	$0.615597373$	1.421661237	\( -\frac{1176039}{11} a + \frac{1096065}{11} \)	\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 322 a - 237\) , \( -1953 a + 298\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(322a-237\right){x}-1953a+298$
53361.3-e1	53361.3-e	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{7} \cdot 7^{10} \cdot 11^{4} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$1$	$0.591437168$	2.731731268	\( \frac{1797450551}{871563} a - \frac{639516755}{79233} \)	\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 7 a + 181\) , \( -1119 a + 741\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a+181\right){x}-1119a+741$
53361.3-e2	53361.3-e	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{8} \cdot 7^{8} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$1.182874337$	2.731731268	\( \frac{332137}{539} a - \frac{2907584}{1617} \)	\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 37 a - 14\) , \( -45 a - 66\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(37a-14\right){x}-45a-66$
53361.3-f1	53361.3-f	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{30} \cdot 7^{6} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{3} \)	$4.324920014$	$0.223801208$	4.470641716	\( \frac{9090072503}{5845851} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 393 a + 651\) , \( -1469 a - 1889\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(393a+651\right){x}-1469a-1889$
53361.3-f2	53361.3-f	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{18} \cdot 7^{6} \cdot 11^{4} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	\( 2^{5} \)	$2.162460007$	$0.447602416$	4.470641716	\( \frac{169112377}{88209} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -102 a - 174\) , \( -479 a - 239\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-102a-174\right){x}-479a-239$
53361.3-f3	53361.3-f	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{12} \cdot 7^{6} \cdot 11^{2} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{3} \)	$1.081230003$	$0.895204832$	4.470641716	\( \frac{30664297}{297} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -57 a - 99\) , \( 271 a + 325\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-57a-99\right){x}+271a+325$
53361.3-f4	53361.3-f	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	53361.3	\( 3^{2} \cdot 7^{2} \cdot 11^{2} \)	\( 3^{12} \cdot 7^{6} \cdot 11^{8} \)	$2.35237$	$(-2a+1), (3a-2), (11)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	\( 2^{3} \)	$4.324920014$	$0.223801208$	4.470641716	\( \frac{347873904937}{395307} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -1317 a - 2199\) , \( -43409 a - 34745\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1317a-2199\right){x}-43409a-34745$

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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.