Elliptic curves over \(\Q(\sqrt{-35}) \) of conductor 576.2

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 35

Note: The completeness Only modular elliptic curves are included

Results (28 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
576.2-a1	576.2-a	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{16} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{2} \)	$1$	$0.908836754$	1.228971599	\( \frac{207646}{6561} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 16\) , \( -180\bigr] \)	${y}^2={x}^3-{x}^2+16{x}-180$
576.2-a2	576.2-a	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{8} \cdot 3^{2} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2 \)	$1$	$7.270694035$	1.228971599	\( \frac{2048}{3} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \)	${y}^2={x}^3-{x}^2+{x}$
576.2-a3	576.2-a	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{4} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{4} \)	$1$	$3.635347017$	1.228971599	\( \frac{35152}{9} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -4\) , \( 4\bigr] \)	${y}^2={x}^3-{x}^2-4{x}+4$
576.2-a4	576.2-a	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{8} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{3} \)	$1$	$1.817673508$	1.228971599	\( \frac{1556068}{81} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -24\) , \( -36\bigr] \)	${y}^2={x}^3-{x}^2-24{x}-36$
576.2-a5	576.2-a	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{2} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$16$	\( 2 \)	$1$	$1.817673508$	1.228971599	\( \frac{28756228}{3} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -64\) , \( 220\bigr] \)	${y}^2={x}^3-{x}^2-64{x}+220$
576.2-a6	576.2-a	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{4} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{2} \)	$1$	$0.908836754$	1.228971599	\( \frac{3065617154}{9} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -384\) , \( -2772\bigr] \)	${y}^2={x}^3-{x}^2-384{x}-2772$
576.2-b1	576.2-b	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{18} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$2.225887286$	$2.828203390$	4.256373951	\( -\frac{33968}{81} a - \frac{883600}{81} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -25 a - 18\) , \( 83 a - 45\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-25a-18\right){x}+83a-45$
576.2-b2	576.2-b	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{24} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$4.451774572$	$1.414101695$	4.256373951	\( \frac{2696540}{6561} a - \frac{7331072}{6561} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -45 a + 162\) , \( 351 a + 1107\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-45a+162\right){x}+351a+1107$
576.2-b3	576.2-b	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{30} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$8.903549145$	$0.707050847$	4.256373951	\( -\frac{48330542590}{43046721} a - \frac{35285554082}{43046721} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 315 a + 162\) , \( 639 a + 12123\bigr] \)	${y}^2={x}^3+a{x}^2+\left(315a+162\right){x}+639a+12123$
576.2-b4	576.2-b	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{24} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$2.225887286$	$0.707050847$	4.256373951	\( -\frac{8920140482}{6561} a + \frac{1140623138}{729} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -725 a + 3042\) , \( 14655 a + 60939\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-725a+3042\right){x}+14655a+60939$
576.2-c1	576.2-c	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{6} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{5} \)	$1.048123068$	$2.828203390$	4.008472266	\( \frac{33968}{81} a - 11328 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a - 1\) , \( -a + 1\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(-2a-1\right){x}-a+1$
576.2-c2	576.2-c	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{18} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{5} \)	$1.048123068$	$0.707050847$	4.008472266	\( \frac{48330542590}{43046721} a - \frac{9290677408}{4782969} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 38 a - 21\) , \( 119 a + 333\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(38a-21\right){x}+119a+333$
576.2-c3	576.2-c	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{12} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{5} \)	$2.096246137$	$1.414101695$	4.008472266	\( -\frac{2696540}{6561} a - \frac{514948}{729} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a - 21\) , \( 15 a + 45\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(-2a-21\right){x}+15a+45$
576.2-c4	576.2-c	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{12} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$4.192492275$	$0.707050847$	4.008472266	\( \frac{8920140482}{6561} a + \frac{1345467760}{6561} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -42 a - 341\) , \( 615 a + 2253\bigr] \)	${y}^2={x}^3+\left(-a-1\right){x}^2+\left(-42a-341\right){x}+615a+2253$
576.2-d1	576.2-d	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{18} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$2.225887286$	$2.828203390$	4.256373951	\( \frac{33968}{81} a - 11328 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 25 a - 43\) , \( -83 a + 38\bigr] \)	${y}^2={x}^3+\left(-a+1\right){x}^2+\left(25a-43\right){x}-83a+38$
576.2-d2	576.2-d	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{30} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$8.903549145$	$0.707050847$	4.256373951	\( \frac{48330542590}{43046721} a - \frac{9290677408}{4782969} \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -315 a + 477\) , \( -639 a + 12762\bigr] \)	${y}^2={x}^3+\left(-a+1\right){x}^2+\left(-315a+477\right){x}-639a+12762$
576.2-d3	576.2-d	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{24} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$4.451774572$	$1.414101695$	4.256373951	\( -\frac{2696540}{6561} a - \frac{514948}{729} \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 45 a + 117\) , \( -351 a + 1458\bigr] \)	${y}^2={x}^3+\left(-a+1\right){x}^2+\left(45a+117\right){x}-351a+1458$
576.2-d4	576.2-d	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{24} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$2.225887286$	$0.707050847$	4.256373951	\( \frac{8920140482}{6561} a + \frac{1345467760}{6561} \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 725 a + 2317\) , \( -14655 a + 75594\bigr] \)	${y}^2={x}^3+\left(-a+1\right){x}^2+\left(725a+2317\right){x}-14655a+75594$
576.2-e1	576.2-e	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{6} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{5} \)	$1.048123068$	$2.828203390$	4.008472266	\( -\frac{33968}{81} a - \frac{883600}{81} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a - 4\) , \( 4 a - 4\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(4a-4\right){x}+4a-4$
576.2-e2	576.2-e	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{12} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{5} \)	$2.096246137$	$1.414101695$	4.008472266	\( \frac{2696540}{6561} a - \frac{7331072}{6561} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a - 24\) , \( -12 a + 36\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(4a-24\right){x}-12a+36$
576.2-e3	576.2-e	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{18} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{5} \)	$1.048123068$	$0.707050847$	4.008472266	\( -\frac{48330542590}{43046721} a - \frac{35285554082}{43046721} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -36 a + 16\) , \( -156 a + 468\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(-36a+16\right){x}-156a+468$
576.2-e4	576.2-e	$4$	$4$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{12} \)	$2.58987$	$(3,a), (3,a+2), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$4.192492275$	$0.707050847$	4.008472266	\( -\frac{8920140482}{6561} a + \frac{1140623138}{729} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 44 a - 384\) , \( -572 a + 2484\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(44a-384\right){x}-572a+2484$
576.2-f1	576.2-f	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{28} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	\( 2^{6} \)	$1$	$0.908836754$	4.915886399	\( \frac{207646}{6561} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 16 a - 144\) , \( -1440 a - 1620\bigr] \)	${y}^2={x}^3+a{x}^2+\left(16a-144\right){x}-1440a-1620$
576.2-f2	576.2-f	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{8} \cdot 3^{14} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2 \)	$1$	$7.270694035$	4.915886399	\( \frac{2048}{3} \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -a - 8\) , \( 0\bigr] \)	${y}^2={x}^3+\left(-a+1\right){x}^2+\left(-a-8\right){x}$
576.2-f3	576.2-f	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{16} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{4} \)	$1$	$3.635347017$	4.915886399	\( \frac{35152}{9} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -4 a + 36\) , \( 32 a + 36\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-4a+36\right){x}+32a+36$
576.2-f4	576.2-f	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{20} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$4$	\( 2^{5} \)	$1$	$1.817673508$	4.915886399	\( \frac{1556068}{81} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -24 a + 216\) , \( -288 a - 324\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-24a+216\right){x}-288a-324$
576.2-f5	576.2-f	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{20} \cdot 3^{14} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$16$	\( 2 \)	$1$	$1.817673508$	4.915886399	\( \frac{28756228}{3} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -64 a + 576\) , \( 1760 a + 1980\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-64a+576\right){x}+1760a+1980$
576.2-f6	576.2-f	$6$	$8$	\(\Q(\sqrt{-35}) \)	$2$	$[0, 1]$	576.2	\( 2^{6} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{16} \)	$2.58987$	$(3,a), (3,a+2), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$16$	\( 2^{2} \)	$1$	$0.908836754$	4.915886399	\( \frac{3065617154}{9} \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -384 a + 3456\) , \( -22176 a - 24948\bigr] \)	${y}^2={x}^3+a{x}^2+\left(-384a+3456\right){x}-22176a-24948$

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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.