Elliptic curves over \(\Q(\sqrt{-71}) \) of conductor 576.11

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 1000 over imaginary quadratic fields with absolute discriminant 71

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
576.11-a1	576.11-a	$6$	$8$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{16} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{2} \)	$4.877020487$	$0.908836754$	2.104123753	\( \frac{207646}{6561} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 16\) , \( -180\bigr] \)	${y}^2={x}^3-{x}^2+16{x}-180$
576.11-a2	576.11-a	$6$	$8$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{8} \cdot 3^{2} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{2} \)	$2.438510243$	$7.270694035$	2.104123753	\( \frac{2048}{3} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \)	${y}^2={x}^3-{x}^2+{x}$
576.11-a3	576.11-a	$6$	$8$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{4} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	$1$	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{6} \)	$1.219255121$	$3.635347017$	2.104123753	\( \frac{35152}{9} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -4\) , \( 4\bigr] \)	${y}^2={x}^3-{x}^2-4{x}+4$
576.11-a4	576.11-a	$6$	$8$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{8} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{4} \)	$2.438510243$	$1.817673508$	2.104123753	\( \frac{1556068}{81} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -24\) , \( -36\bigr] \)	${y}^2={x}^3-{x}^2-24{x}-36$
576.11-a5	576.11-a	$6$	$8$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{2} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{2} \)	$2.438510243$	$1.817673508$	2.104123753	\( \frac{28756228}{3} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -64\) , \( 220\bigr] \)	${y}^2={x}^3-{x}^2-64{x}+220$
576.11-a6	576.11-a	$6$	$8$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{4} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{2} \)	$4.877020487$	$0.908836754$	2.104123753	\( \frac{3065617154}{9} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -384\) , \( -2772\bigr] \)	${y}^2={x}^3-{x}^2-384{x}-2772$
576.11-b1	576.11-b	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{14} \cdot 3^{4} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{2} \)	$1$	$2.604826897$	0.618272157	\( \frac{7464940}{27} a - \frac{22374088}{27} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a - 15\) , \( -4 a - 18\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(-4a-15\right){x}-4a-18$
576.11-b2	576.11-b	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{28} \cdot 3^{8} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{4} \)	$1$	$2.604826897$	0.618272157	\( \frac{147200}{729} a + \frac{943312}{729} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a - 20\) , \( 0\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(-4a-20\right){x}$
576.11-b3	576.11-b	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{32} \cdot 3^{13} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{3} \)	$1$	$1.302413448$	0.618272157	\( -\frac{122691544}{531441} a + \frac{932972164}{531441} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 16 a + 80\) , \( -112 a + 208\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(16a+80\right){x}-112a+208$
576.11-b4	576.11-b	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{14} \cdot 3^{7} \cdot 5^{12} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{4} \)	$1$	$2.604826897$	0.618272157	\( -\frac{1083116}{81} a + \frac{107020}{27} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -82 a + 132\) , \( -235 a + 2034\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(-82a+132\right){x}-235a+2034$
576.11-c1	576.11-c	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{14} \cdot 3^{4} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{2} \)	$1$	$2.604826897$	0.618272157	\( -\frac{7464940}{27} a - \frac{1656572}{3} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 6 a - 20\) , \( 9 a - 42\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(6a-20\right){x}+9a-42$
576.11-c2	576.11-c	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{14} \cdot 3^{7} \cdot 5^{12} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{4} \)	$1$	$2.604826897$	0.618272157	\( \frac{1083116}{81} a - \frac{762056}{81} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 84 a + 49\) , \( 318 a + 1848\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(84a+49\right){x}+318a+1848$
576.11-c3	576.11-c	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{28} \cdot 3^{8} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{4} \)	$1$	$2.604826897$	0.618272157	\( -\frac{147200}{729} a + \frac{121168}{81} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 6 a - 25\) , \( 5 a - 25\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(6a-25\right){x}+5a-25$
576.11-c4	576.11-c	$4$	$4$	\(\Q(\sqrt{-71}) \)	$2$	$[0, 1]$	576.11	\( 2^{6} \cdot 3^{2} \)	\( 2^{32} \cdot 3^{13} \)	$3.68870$	$(2,a), (2,a+1), (3,a), (3,a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{3} \)	$1$	$1.302413448$	0.618272157	\( \frac{122691544}{531441} a + \frac{90031180}{59049} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -14 a + 95\) , \( 97 a + 191\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(-14a+95\right){x}+97a+191$

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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.