Elliptic curves over \(\Q(\sqrt{10}) \) of conductor 256.1

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 (16 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
256.1-a1	256.1-a	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{16} \)	$2.26063$	$(2,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$2.442170565$	$7.547952572$	5.829148982	\( 128 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( -1\bigr] \)	${y}^2={x}^{3}-{x}^{2}+{x}-1$
256.1-a2	256.1-a	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{26} \)	$2.26063$	$(2,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$1.221085282$	$15.09590514$	5.829148982	\( 10976 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -9\) , \( -7\bigr] \)	${y}^2={x}^{3}-{x}^{2}-9{x}-7$
256.1-b1	256.1-b	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{12} \)	$2.26063$	$(2,a)$	$2$	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 1 \)	$1.838697222$	$13.75037163$	3.997556959	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -6 a + 19\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(-6a+19\right){x}$
256.1-b2	256.1-b	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{24} \)	$2.26063$	$(2,a)$	$2$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓	$2$	2Cs	$1$	\( 2 \)	$1.838697222$	$27.50074327$	3.997556959	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 24 a - 76\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(24a-76\right){x}$
256.1-b3	256.1-b	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{30} \)	$2.26063$	$(2,a)$	$2$	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2^{2} \)	$0.459674305$	$13.75037163$	3.997556959	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 264 a - 836\) , \( 4144 a - 13104\bigr] \)	${y}^2={x}^{3}+\left(264a-836\right){x}+4144a-13104$
256.1-b4	256.1-b	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{30} \)	$2.26063$	$(2,a)$	$2$	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2^{2} \)	$0.459674305$	$13.75037163$	3.997556959	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 264 a - 836\) , \( -4144 a + 13104\bigr] \)	${y}^2={x}^{3}+\left(264a-836\right){x}-4144a+13104$
256.1-c1	256.1-c	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{16} \)	$2.26063$	$(2,a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2 \)	$0.452304684$	$16.29302268$	2.330412212	\( 128 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 1\bigr] \)	${y}^2={x}^{3}+{x}^{2}+{x}+1$
256.1-c2	256.1-c	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{26} \)	$2.26063$	$(2,a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$0.113076171$	$32.58604536$	2.330412212	\( 10976 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -9\) , \( 7\bigr] \)	${y}^2={x}^{3}+{x}^{2}-9{x}+7$
256.1-d1	256.1-d	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{28} \)	$2.26063$	$(2,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$4$	\( 2 \)	$1$	$16.29302268$	5.152306164	\( 128 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( 3\) , \( 5\bigr] \)	${y}^2={x}^{3}-{x}^{2}+3{x}+5$
256.1-d2	256.1-d	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{14} \)	$2.26063$	$(2,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$32.58604536$	5.152306164	\( 10976 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -2\) , \( 2\bigr] \)	${y}^2={x}^{3}-{x}^{2}-2{x}+2$
256.1-e1	256.1-e	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{24} \)	$2.26063$	$(2,a)$	0	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓			$4$	\( 1 \)	$1$	$13.75037163$	2.174124652	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -24 a + 76\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(-24a+76\right){x}$
256.1-e2	256.1-e	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{12} \)	$2.26063$	$(2,a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓	$2$	2Cs	$4$	\( 2 \)	$1$	$27.50074327$	2.174124652	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 6 a - 19\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(6a-19\right){x}$
256.1-e3	256.1-e	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{18} \)	$2.26063$	$(2,a)$	0	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2^{2} \)	$1$	$13.75037163$	2.174124652	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 66 a - 209\) , \( 518 a - 1638\bigr] \)	${y}^2={x}^{3}+\left(66a-209\right){x}+518a-1638$
256.1-e4	256.1-e	$4$	$4$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{18} \)	$2.26063$	$(2,a)$	0	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2^{2} \)	$1$	$13.75037163$	2.174124652	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 66 a - 209\) , \( -518 a + 1638\bigr] \)	${y}^2={x}^{3}+\left(66a-209\right){x}-518a+1638$
256.1-f1	256.1-f	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{28} \)	$2.26063$	$(2,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$0.843022143$	$7.547952572$	2.012186100	\( 128 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( 3\) , \( -5\bigr] \)	${y}^2={x}^{3}+{x}^{2}+3{x}-5$
256.1-f2	256.1-f	$2$	$2$	\(\Q(\sqrt{10}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{14} \)	$2.26063$	$(2,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$0.421511071$	$15.09590514$	2.012186100	\( 10976 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -2\) , \( -2\bigr] \)	${y}^2={x}^{3}+{x}^{2}-2{x}-2$

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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.