Elliptic curves over Q(−2)\Q(\sqrt{-2}) of conductor 25992.5

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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
25992.5-a1	25992.5-a	$1$	$1$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{4} \cdot 19^{6}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓					$1$	$2^{3} \cdot 3^{2}$	$0.014048452$	$1.742877918$	4.986237382	$\frac{70575104}{61731}$	$\bigl[0$ , $-1$ , $a$ , $14$ , $-18\bigr]$	${y}^2+a{y}={x}^{3}-{x}^{2}+14{x}-18$
25992.5-b1	25992.5-b	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{11} \cdot 3^{5} \cdot 19^{5}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{3}$	$1.113314158$	$1.016839116$	3.201953129	$-\frac{911099210395}{10556001} a - \frac{662663918104}{10556001}$	$\bigl[a$ , $1$ , $a$ , $50 a + 87$ , $174 a - 373\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(50a+87\right){x}+174a-373$
25992.5-b2	25992.5-b	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{11} \cdot 3^{5} \cdot 19^{5}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{3}$	$1.113314158$	$1.016839116$	3.201953129	$\frac{911099210395}{10556001} a - \frac{662663918104}{10556001}$	$\bigl[a$ , $1$ , $a$ , $-50 a + 87$ , $-174 a - 373\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-50a+87\right){x}-174a-373$
25992.5-b3	25992.5-b	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{10} \cdot 3^{4} \cdot 19^{4}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	$2^{5}$	$0.556657079$	$2.033678232$	3.201953129	$\frac{715822}{3249}$	$\bigl[a$ , $1$ , $a$ , $7$ , $-13\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+7{x}-13$
25992.5-b4	25992.5-b	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{2} \cdot 19^{2}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2$	$1.113314158$	$4.067356465$	3.201953129	$\frac{470596}{57}$	$\bigl[a$ , $1$ , $a$ , $-3$ , $-3\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-3{x}-3$
25992.5-c1	25992.5-c	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{3} \cdot 19^{3}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$1.337629533$	0.945846913	$-\frac{108662362112}{3249} a - \frac{18202716160}{3249}$	$\bigl[0$ , $-1$ , $0$ , $76 a - 95$ , $422 a - 166\bigr]$	${y}^2={x}^{3}-{x}^{2}+\left(76a-95\right){x}+422a-166$
25992.5-c2	25992.5-c	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{10} \cdot 3^{6} \cdot 19^{9}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$0.334407383$	0.945846913	$-\frac{410003128905041890}{1375668606321} a - \frac{68512985349777374}{1375668606321}$	$\bigl[a$ , $1$ , $0$ , $79 a + 1230$ , $-11696 a + 2435\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(79a+1230\right){x}-11696a+2435$
25992.5-c3	25992.5-c	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{10} \cdot 3^{24} \cdot 19^{3}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$0.334407383$	0.945846913	$\frac{354192680084050}{15539866281} a - \frac{588677928726082}{15539866281}$	$\bigl[a$ , $1$ , $0$ , $-581 a + 250$ , $-2356 a + 9323\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-581a+250\right){x}-2356a+9323$
25992.5-c4	25992.5-c	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{12} \cdot 19^{6}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{5}$	$1$	$0.668814766$	0.945846913	$\frac{187305362200}{855036081} a + \frac{91888723868}{855036081}$	$\bigl[a$ , $1$ , $0$ , $-11 a + 60$ , $-266 a + 203\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-11a+60\right){x}-266a+203$
25992.5-c5	25992.5-c	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{6} \cdot 19^{6}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{6}$	$1$	$1.337629533$	0.945846913	$-\frac{38345975360}{10556001} a + \frac{56441533136}{10556001}$	$\bigl[a$ , $1$ , $0$ , $19 a - 25$ , $-51 a + 15\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(19a-25\right){x}-51a+15$
25992.5-c6	25992.5-c	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{3} \cdot 19^{9}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{5}$	$1$	$0.668814766$	0.945846913	$\frac{562622763686312}{152852067369} a + \frac{824822139136660}{152852067369}$	$\bigl[a$ , $1$ , $0$ , $49 a - 130$ , $276 a - 531\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(49a-130\right){x}+276a-531$
25992.5-d1	25992.5-d	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{3} \cdot 19^{3}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$1.337629533$	0.945846913	$\frac{108662362112}{3249} a - \frac{18202716160}{3249}$	$\bigl[0$ , $-1$ , $0$ , $-76 a - 95$ , $-422 a - 166\bigr]$	${y}^2={x}^{3}-{x}^{2}+\left(-76a-95\right){x}-422a-166$
25992.5-d2	25992.5-d	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{10} \cdot 3^{6} \cdot 19^{9}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$0.334407383$	0.945846913	$\frac{410003128905041890}{1375668606321} a - \frac{68512985349777374}{1375668606321}$	$\bigl[a$ , $1$ , $0$ , $-79 a + 1230$ , $11696 a + 2435\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-79a+1230\right){x}+11696a+2435$
25992.5-d3	25992.5-d	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{10} \cdot 3^{24} \cdot 19^{3}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$0.334407383$	0.945846913	$-\frac{354192680084050}{15539866281} a - \frac{588677928726082}{15539866281}$	$\bigl[a$ , $1$ , $0$ , $581 a + 250$ , $2356 a + 9323\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(581a+250\right){x}+2356a+9323$
25992.5-d4	25992.5-d	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{12} \cdot 19^{6}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{5}$	$1$	$0.668814766$	0.945846913	$-\frac{187305362200}{855036081} a + \frac{91888723868}{855036081}$	$\bigl[a$ , $1$ , $0$ , $11 a + 60$ , $266 a + 203\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(11a+60\right){x}+266a+203$
25992.5-d5	25992.5-d	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{6} \cdot 19^{6}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{6}$	$1$	$1.337629533$	0.945846913	$\frac{38345975360}{10556001} a + \frac{56441533136}{10556001}$	$\bigl[a$ , $1$ , $0$ , $-19 a - 25$ , $51 a + 15\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-19a-25\right){x}+51a+15$
25992.5-d6	25992.5-d	$6$	$8$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{3} \cdot 19^{9}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{5}$	$1$	$0.668814766$	0.945846913	$-\frac{562622763686312}{152852067369} a + \frac{824822139136660}{152852067369}$	$\bigl[a$ , $1$ , $0$ , $-49 a - 130$ , $-276 a - 531\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-49a-130\right){x}-276a-531$
25992.5-e1	25992.5-e	$2$	$2$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{12} \cdot 19^{4}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3} \cdot 3$	$1$	$0.673222874$	2.856242760	$-\frac{2491909313363360}{405017091} a - \frac{3136318197436624}{405017091}$	$\bigl[a$ , $a - 1$ , $a$ , $-154 a + 430$ , $2410 a + 2502\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-154a+430\right){x}+2410a+2502$
25992.5-e2	25992.5-e	$2$	$2$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{6} \cdot 19^{8}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3} \cdot 3$	$1$	$0.673222874$	2.856242760	$\frac{24757755545600}{11432149083} a - \frac{7750500325376}{11432149083}$	$\bigl[0$ , $a$ , $0$ , $-42 a + 100$ , $-288 a - 267\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-42a+100\right){x}-288a-267$
25992.5-f1	25992.5-f	$1$	$1$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{12} \cdot 19^{2}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2^{5}$	$0.063609970$	$1.735435969$	4.995727764	$-\frac{7265024000}{124659} a + \frac{4315648000}{124659}$	$\bigl[0$ , $a$ , $a$ , $-25 a - 4$ , $45 a - 31\bigr]$	${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(-25a-4\right){x}+45a-31$
25992.5-g1	25992.5-g	$1$	$1$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{12} \cdot 19^{2}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2^{5}$	$0.063609970$	$1.735435969$	4.995727764	$\frac{7265024000}{124659} a + \frac{4315648000}{124659}$	$\bigl[0$ , $-a$ , $a$ , $25 a - 4$ , $-45 a - 31\bigr]$	${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(25a-4\right){x}-45a-31$
25992.5-h1	25992.5-h	$2$	$2$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{12} \cdot 19^{4}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3} \cdot 3$	$1$	$0.673222874$	2.856242760	$\frac{2491909313363360}{405017091} a - \frac{3136318197436624}{405017091}$	$\bigl[a$ , $-a - 1$ , $a$ , $154 a + 430$ , $-2410 a + 2502\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(154a+430\right){x}-2410a+2502$
25992.5-h2	25992.5-h	$2$	$2$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{6} \cdot 19^{8}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3} \cdot 3$	$1$	$0.673222874$	2.856242760	$-\frac{24757755545600}{11432149083} a - \frac{7750500325376}{11432149083}$	$\bigl[0$ , $-a$ , $0$ , $42 a + 100$ , $288 a - 267\bigr]$	${y}^2={x}^{3}-a{x}^{2}+\left(42a+100\right){x}+288a-267$
25992.5-i1	25992.5-i	$1$	$1$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{12} \cdot 19^{2}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓					$1$	$2^{3} \cdot 3^{2}$	$0.032122221$	$2.067833808$	6.763456351	$-\frac{81415168}{13851}$	$\bigl[0$ , $1$ , $a$ , $-14$ , $-28\bigr]$	${y}^2+a{y}={x}^{3}+{x}^{2}-14{x}-28$
25992.5-j1	25992.5-j	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{10} \cdot 3^{24} \cdot 19^{2}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{5} \cdot 3^{2}$	$0.549034424$	$0.527684085$	7.374983868	$\frac{9878111854}{10097379}$	$\bigl[a$ , $0$ , $0$ , $142$ , $546\bigr]$	${y}^2+a{x}{y}={x}^{3}+142{x}+546$
25992.5-j2	25992.5-j	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{8} \cdot 3^{12} \cdot 19^{4}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	$2^{5} \cdot 3^{2}$	$0.274517212$	$1.055368171$	7.374983868	$\frac{768400132}{263169}$	$\bigl[a$ , $0$ , $0$ , $-48$ , $90\bigr]$	${y}^2+a{x}{y}={x}^{3}-48{x}+90$
25992.5-j3	25992.5-j	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{10} \cdot 3^{6} \cdot 19^{8}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3} \cdot 3^{2}$	$0.549034424$	$0.527684085$	7.374983868	$\frac{111223479026}{3518667}$	$\bigl[a$ , $0$ , $0$ , $-318$ , $-2070\bigr]$	${y}^2+a{x}{y}={x}^{3}-318{x}-2070$
25992.5-j4	25992.5-j	$4$	$4$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	25992.5	$2^{3} \cdot 3^{2} \cdot 19^{2}$	$2^{4} \cdot 3^{6} \cdot 19^{2}$	$3.20918$	$(a), (-a-1), (a-1), (-3a+1), (3a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2 \cdot 3^{2}$	$0.549034424$	$2.110736342$	7.374983868	$\frac{2211014608}{513}$	$\bigl[a$ , $0$ , $0$ , $-43$ , $116\bigr]$	${y}^2+a{x}{y}={x}^{3}-43{x}+116$

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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.