Elliptic curves over \(\Q(\sqrt{-7}) \) of conductor 10404.2

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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
10404.2-a1	10404.2-a	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{2} \cdot 3^{8} \cdot 17^{4} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{3} \)	$0.286507785$	$2.302301329$	1.994525346	\( \frac{46268279}{46818} \)	\( \bigl[1\) , \( 1\) , \( 0\) , \( 8\) , \( 10\bigr] \)	${y}^2+{x}{y}={x}^{3}+{x}^{2}+8{x}+10$
10404.2-a2	10404.2-a	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{4} \cdot 3^{4} \cdot 17^{2} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{3} \)	$0.143253892$	$4.604602658$	1.994525346	\( \frac{1771561}{612} \)	\( \bigl[1\) , \( 1\) , \( 0\) , \( -2\) , \( 0\bigr] \)	${y}^2+{x}{y}={x}^{3}+{x}^{2}-2{x}$
10404.2-b1	10404.2-b	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{6} \cdot 3^{24} \cdot 17^{4} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{3} \cdot 3 \)	$5.231471366$	$0.415867934$	2.192799889	\( -\frac{1107111813625}{1228691592} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -216\) , \( 2062\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-216{x}+2062$
10404.2-b2	10404.2-b	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{18} \cdot 3^{8} \cdot 17^{12} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{3} \cdot 3 \)	$1.743823788$	$0.138622644$	2.192799889	\( \frac{655215969476375}{1001033261568} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( 1809\) , \( -37790\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+1809{x}-37790$
10404.2-b3	10404.2-b	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{36} \cdot 3^{4} \cdot 17^{6} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{3} \cdot 3 \)	$0.871911894$	$0.277245289$	2.192799889	\( \frac{46753267515625}{11591221248} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -751\) , \( -6046\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-751{x}-6046$
10404.2-b4	10404.2-b	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{12} \cdot 3^{12} \cdot 17^{2} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{3} \cdot 3 \)	$2.615735683$	$0.831735869$	2.192799889	\( \frac{1845026709625}{793152} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -256\) , \( 1550\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-256{x}+1550$
10404.2-c1	10404.2-c	$6$	$8$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{2} \cdot 3^{4} \cdot 17^{16} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$4$	\( 2^{4} \)	$1.755122514$	$0.183754147$	7.801453798	\( -\frac{491411892194497}{125563633938} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -1644\) , \( -30942\bigr] \)	${y}^2+{x}{y}={x}^{3}-1644{x}-30942$
10404.2-c2	10404.2-c	$6$	$8$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{4} \cdot 3^{32} \cdot 17^{2} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{6} \)	$3.510245028$	$0.367508294$	7.801453798	\( \frac{1276229915423}{2927177028} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( 226\) , \( -2232\bigr] \)	${y}^2+{x}{y}={x}^{3}+226{x}-2232$
10404.2-c3	10404.2-c	$6$	$8$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{8} \cdot 3^{16} \cdot 17^{4} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{8} \)	$1.755122514$	$0.735016588$	7.801453798	\( \frac{163936758817}{30338064} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -114\) , \( -396\bigr] \)	${y}^2+{x}{y}={x}^{3}-114{x}-396$
10404.2-c4	10404.2-c	$6$	$8$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{16} \cdot 3^{8} \cdot 17^{2} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{8} \)	$0.877561257$	$1.470033177$	7.801453798	\( \frac{4354703137}{352512} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( 68\bigr] \)	${y}^2+{x}{y}={x}^{3}-34{x}+68$
10404.2-c5	10404.2-c	$6$	$8$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{4} \cdot 3^{8} \cdot 17^{8} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{6} \)	$0.877561257$	$0.367508294$	7.801453798	\( \frac{576615941610337}{27060804} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -1734\) , \( -27936\bigr] \)	${y}^2+{x}{y}={x}^{3}-1734{x}-27936$
10404.2-c6	10404.2-c	$6$	$8$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	10404.2	\( 2^{2} \cdot 3^{2} \cdot 17^{2} \)	\( 2^{2} \cdot 3^{4} \cdot 17^{4} \)	$2.38774$	$(a), (-a+1), (3), (17)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$16$	\( 2^{2} \)	$1.755122514$	$0.183754147$	7.801453798	\( \frac{2361739090258884097}{5202} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -27744\) , \( -1781010\bigr] \)	${y}^2+{x}{y}={x}^{3}-27744{x}-1781010$

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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.