Elliptic curves over \(\Q(\sqrt{-7}) \) of conductor 2048.6

Results (20 matches)

 

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
2048.6-a1	2048.6-a	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{25} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2 \)	$1.396505574$	$2.178762637$	2.300030359	\( 237572 a - 523524 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -10 a - 21\) , \( -47 a - 29\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-10a-21\right){x}-47a-29$
2048.6-a2	2048.6-a	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{28} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2^{3} \)	$0.349126393$	$2.178762637$	2.300030359	\( -3084 a - 62716 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -15 a + 9\) , \( 11 a - 25\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-15a+9\right){x}+11a-25$
2048.6-a3	2048.6-a	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{20} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2Cs	$1$	\( 2^{3} \)	$0.698252787$	$4.357525275$	2.300030359	\( -336 a + 560 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -1\) , \( -a - 1\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}-{x}-a-1$
2048.6-a4	2048.6-a	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{19} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2^{2} \)	$0.349126393$	$4.357525275$	2.300030359	\( 2056 a + 2768 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( a - 3\) , \( a - 3\bigr] \)	${y}^2={x}^{3}+{x}^{2}+\left(a-3\right){x}+a-3$
2048.6-b1	2048.6-b	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{18} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2Cs, 3Nn	$1$	\( 2^{3} \)	$0.657095919$	$4.941210318$	2.454387246	\( -64 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -1\) , \( a - 1\bigr] \)	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}-{x}+a-1$
2048.6-b2	2048.6-b	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{15} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2B, 3Nn	$1$	\( 1 \)	$1.314191839$	$4.941210318$	2.454387246	\( -17416 a + 20208 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -a + 3\) , \( 2 a + 2\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(-a+3\right){x}+2a+2$
2048.6-b3	2048.6-b	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{27} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2B, 3Nn	$1$	\( 2^{3} \)	$0.328547959$	$2.470605159$	2.454387246	\( 17416 a + 2792 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 5 a + 9\) , \( 7 a - 13\bigr] \)	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(5a+9\right){x}+7a-13$
2048.6-b4	2048.6-b	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{24} \)	$1.59045$	$(a), (-a+1)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2B, 3Nn	$1$	\( 2^{3} \)	$1.314191839$	$2.470605159$	2.454387246	\( 238328 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 10 a - 21\) , \( 31 a - 29\bigr] \)	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(10a-21\right){x}+31a-29$
2048.6-c1	2048.6-c	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{18} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2Cs, 3Nn	$1$	\( 2^{3} \)	$1$	$4.941210318$	1.867601953	\( -64 \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -1\) , \( -a + 1\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}-{x}-a+1$
2048.6-c2	2048.6-c	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{15} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2B, 3Nn	$1$	\( 2 \)	$1$	$4.941210318$	1.867601953	\( -17416 a + 20208 \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( -a + 3\) , \( -2 a - 2\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+\left(-a+3\right){x}-2a-2$
2048.6-c3	2048.6-c	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{27} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2B, 3Nn	$1$	\( 2^{4} \)	$1$	$2.470605159$	1.867601953	\( 17416 a + 2792 \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 5 a + 9\) , \( -7 a + 13\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(5a+9\right){x}-7a+13$
2048.6-c4	2048.6-c	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{24} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2B, 3Nn	$1$	\( 2^{2} \)	$1$	$2.470605159$	1.867601953	\( 238328 \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 10 a - 21\) , \( -31 a + 29\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(10a-21\right){x}-31a+29$
2048.6-d1	2048.6-d	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{18} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2 \)	$1$	$4.861490513$	1.837470700	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -a - 1\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}$
2048.6-d2	2048.6-d	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{18} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓			✓	$2$	2Cs	$1$	\( 2^{3} \)	$1$	$4.861490513$	1.837470700	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -a + 2\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(-a+2\right){x}$
2048.6-d3	2048.6-d	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{24} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2^{2} \)	$1$	$2.430745256$	1.837470700	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -11 a + 22\) , \( -14 a - 28\bigr] \)	${y}^2={x}^{3}+\left(-11a+22\right){x}-14a-28$
2048.6-d4	2048.6-d	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{24} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓			✓			$1$	\( 2^{2} \)	$1$	$2.430745256$	1.837470700	\( 287496 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -11 a + 22\) , \( 14 a + 28\bigr] \)	${y}^2={x}^{3}+\left(-11a+22\right){x}+14a+28$
2048.6-e1	2048.6-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{25} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$2.178762637$	1.646989744	\( 237572 a - 523524 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -10 a - 21\) , \( 47 a + 29\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-10a-21\right){x}+47a+29$
2048.6-e2	2048.6-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{28} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$2.178762637$	1.646989744	\( -3084 a - 62716 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -15 a + 9\) , \( -11 a + 25\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15a+9\right){x}-11a+25$
2048.6-e3	2048.6-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{20} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2Cs	$1$	\( 2^{3} \)	$1$	$4.357525275$	1.646989744	\( -336 a + 560 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1\) , \( a + 1\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-{x}+a+1$
2048.6-e4	2048.6-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	2048.6	\( 2^{11} \)	\( 2^{19} \)	$1.59045$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2 \)	$1$	$4.357525275$	1.646989744	\( 2056 a + 2768 \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( a - 3\) , \( -a + 3\bigr] \)	${y}^2={x}^{3}-{x}^{2}+\left(a-3\right){x}-a+3$

 

  *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.