Elliptic curves over \(\Q(\sqrt{13}) \) of conductor 1024.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 5000 over real quadratic fields with discriminant 497

Results (20 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
1024.1-a1	1024.1-a	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{12} \)	$1.82257$	$(2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓	$2$	2Cs	$1$	\( 2 \)	$1$	$27.50074327$	0.953416730	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 3 a - 7\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(3a-7\right){x}$
1024.1-a2	1024.1-a	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{24} \)	$1.82257$	$(2)$	0	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2 \)	$1$	$6.875185818$	0.953416730	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -12 a + 28\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(-12a+28\right){x}$
1024.1-a3	1024.1-a	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	0	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 1 \)	$1$	$13.75037163$	0.953416730	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -33 a - 44\) , \( -140 a - 182\bigr] \)	${y}^2={x}^{3}+\left(-33a-44\right){x}-140a-182$
1024.1-a4	1024.1-a	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	0	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 1 \)	$1$	$13.75037163$	0.953416730	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -33 a - 44\) , \( 140 a + 182\bigr] \)	${y}^2={x}^{3}+\left(-33a-44\right){x}+140a+182$
1024.1-b1	1024.1-b	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$3$	3Nn	$1$	\( 1 \)	$1$	$7.681576726$	2.130486058	\( -74088 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( -4 a + 2\bigr] \)	${y}^2={x}^{3}-7{x}-4a+2$
1024.1-c1	1024.1-c	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 2 \)	$0.295484908$	$11.29241653$	3.701779188	\( 264 a - 608 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( a - 2\) , \( -a + 2\bigr] \)	${y}^2={x}^{3}+{x}^{2}+\left(a-2\right){x}-a+2$
1024.1-d1	1024.1-d	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 2 \)	$0.201887458$	$8.453314465$	1.893322871	\( 264 a - 608 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( a - 2\) , \( a - 2\bigr] \)	${y}^2={x}^{3}-{x}^{2}+\left(a-2\right){x}+a-2$
1024.1-e1	1024.1-e	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 2 \)	$0.295484908$	$11.29241653$	3.701779188	\( -264 a - 344 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -a - 1\) , \( a + 1\bigr] \)	${y}^2={x}^{3}+{x}^{2}+\left(-a-1\right){x}+a+1$
1024.1-f1	1024.1-f	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 2 \)	$0.201887458$	$8.453314465$	1.893322871	\( -264 a - 344 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -a - 1\) , \( -a - 1\bigr] \)	${y}^2={x}^{3}-{x}^{2}+\left(-a-1\right){x}-a-1$
1024.1-g1	1024.1-g	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 2 \)	$0.124219855$	$19.81751393$	2.731042807	\( -264 a - 344 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 1\) , \( -3 a + 8\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}-3a+8$
1024.1-h1	1024.1-h	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 2 \)	$0.124219855$	$19.81751393$	2.731042807	\( 264 a - 608 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 1\) , \( 3 a + 4\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+1\right){x}+3a+4$
1024.1-i1	1024.1-i	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$3$	3Nn	$9$	\( 1 \)	$1$	$2.150268075$	5.367393556	\( -74088 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 21 a - 49\) , \( 72 a - 166\bigr] \)	${y}^2={x}^{3}+\left(21a-49\right){x}+72a-166$
1024.1-j1	1024.1-j	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$3$	3Nn	$1$	\( 2^{2} \)	$0.013226887$	$27.44151841$	1.610697971	\( -74088 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 21 a - 49\) , \( -72 a + 166\bigr] \)	${y}^2={x}^{3}+\left(21a-49\right){x}-72a+166$
1024.1-k1	1024.1-k	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$3$	3Nn	$1$	\( 1 \)	$1$	$7.681576726$	2.130486058	\( -74088 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( 4 a - 2\bigr] \)	${y}^2={x}^{3}-7{x}+4a-2$
1024.1-l1	1024.1-l	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{12} \)	$1.82257$	$(2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	\( 2 \)	$1.458127934$	$27.50074327$	2.780407135	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \)	${y}^2={x}^{3}-{x}$
1024.1-l2	1024.1-l	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{24} \)	$1.82257$	$(2)$	$1$	$\Z/4\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{2} \)	$2.916255868$	$6.875185818$	2.780407135	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 4\) , \( 0\bigr] \)	${y}^2={x}^{3}+4{x}$
1024.1-l3	1024.1-l	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 1 \)	$2.916255868$	$6.875185818$	2.780407135	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( -14\bigr] \)	${y}^2={x}^{3}-11{x}-14$
1024.1-l4	1024.1-l	$4$	$4$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	$1$	$\Z/4\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{2} \)	$0.729063967$	$27.50074327$	2.780407135	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( 14\bigr] \)	${y}^2={x}^{3}-11{x}+14$
1024.1-m1	1024.1-m	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 1 \)	$1$	$4.816867964$	1.335958802	\( -264 a - 344 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 1\) , \( 3 a - 8\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+1\right){x}+3a-8$
1024.1-n1	1024.1-n	$1$	$1$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	1024.1	\( 2^{10} \)	\( 2^{18} \)	$1.82257$	$(2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$				✓			$1$	\( 1 \)	$1$	$4.816867964$	1.335958802	\( 264 a - 608 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 1\) , \( -3 a - 4\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}-3a-4$

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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.