Elliptic curves over Q(2)\Q(\sqrt{2}) of conductor 686.1

Results (24 matches)

  Download displayed columns for results to

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
686.1-a1	686.1-a	$2$	$2$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{10} \cdot 7^{10}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2} \cdot 5$	$1$	$1.136251981$	2.008628703	$\frac{2968038}{7} a - \frac{133931057}{224}$	$\bigl[1$ , $-a - 1$ , $a + 1$ , $43 a - 6$ , $65 a - 135\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(43a-6\right){x}+65a-135$
686.1-a2	686.1-a	$2$	$2$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{5} \cdot 7^{11}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2} \cdot 5$	$1$	$1.136251981$	2.008628703	$-\frac{199003633361573}{392} a + \frac{35179204742368}{49}$	$\bigl[1$ , $-a - 1$ , $a + 1$ , $303 a - 646$ , $4929 a - 6831\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(303a-646\right){x}+4929a-6831$
686.1-b1	686.1-b	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{2} \cdot 7^{7}$	$1.29349$	$(a), (-2a+1), (2a+1)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2 \cdot 3^{2}$	$0.050461553$	$20.34834318$	1.452127213	$\frac{7974621}{686} a - \frac{12367339}{686}$	$\bigl[a + 1$ , $a - 1$ , $a$ , $5 a - 9$ , $-15 a + 21\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a-9\right){x}-15a+21$
686.1-b2	686.1-b	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{6} \cdot 7^{13}$	$1.29349$	$(a), (-2a+1), (2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2 \cdot 3^{3}$	$0.016820517$	$2.260927020$	1.452127213	$-\frac{59405903367}{322828856} a + \frac{391232728667}{322828856}$	$\bigl[a + 1$ , $a - 1$ , $a$ , $-25 a + 46$ , $-20 a + 22\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-25a+46\right){x}-20a+22$
686.1-c1	686.1-c	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{2} \cdot 7^{5}$	$1.29349$	$(a), (-2a+1), (2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	$2$	$0.125216197$	$8.219192181$	1.455474647	$-\frac{341511377481251}{686} a - \frac{482970021761709}{686}$	$\bigl[1$ , $a$ , $a + 1$ , $-31 a - 50$ , $121 a + 124\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-31a-50\right){x}+121a+124$
686.1-c2	686.1-c	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{6} \cdot 7^{3}$	$1.29349$	$(a), (-2a+1), (2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	$2$	$0.041738732$	$24.65757654$	1.455474647	$-\frac{208183}{56} a - \frac{292563}{56}$	$\bigl[1$ , $a$ , $a + 1$ , $-a$ , $-a\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}-a{x}-a$
686.1-d1	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{36} \cdot 7^{8}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	$2^{4} \cdot 3^{2}$	$1$	$0.661753152$	2.105685636	$-\frac{548347731625}{1835008}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-683 a - 1536$ , $19222 a + 21843\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-683a-1536\right){x}+19222a+21843$
686.1-d2	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{4} \cdot 7^{8}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	$2^{4}$	$1$	$5.955778371$	2.105685636	$-\frac{15625}{28}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-3 a - 6$ , $-6 a - 7\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a-6\right){x}-6a-7$
686.1-d3	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{12} \cdot 7^{12}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3Cs	$1$	$2^{4} \cdot 3$	$1$	$1.985259457$	2.105685636	$\frac{9938375}{21952}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $17 a + 39$ , $126 a + 143\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(17a+39\right){x}+126a+143$
686.1-d4	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2 \cdot 7^{11}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	$2^{2}$	$1$	$5.955778371$	2.105685636	$-\frac{2928743223192875}{4802} a + \frac{2070934198465000}{2401}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $132 a - 376$ , $-1908 a + 2353\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(132a-376\right){x}-1908a+2353$
686.1-d5	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{6} \cdot 7^{18}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2Cs, 3Cs	$1$	$2^{4} \cdot 3$	$1$	$1.985259457$	2.105685636	$\frac{4956477625}{941192}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-143 a - 321$ , $1534 a + 1743\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-143a-321\right){x}+1534a+1743$
686.1-d6	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{3} \cdot 7^{21}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3Cs	$4$	$2^{2} \cdot 3$	$1$	$0.496314864$	2.105685636	$-\frac{392127492092318125}{55365148804} a + \frac{138814532776321000}{13841287201}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-253 a - 2161$ , $-5170 a - 37057\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-253a-2161\right){x}-5170a-37057$
686.1-d7	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{9} \cdot 7^{11}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$4$	$2^{2} \cdot 3^{2}$	$1$	$0.165438288$	2.105685636	$-\frac{218768831290078842857759125}{76832} a + \frac{9668307757341907702465000}{2401}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $37477 a - 93056$ , $8696054 a - 9208877\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(37477a-93056\right){x}+8696054a-9208877$
686.1-d8	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{2} \cdot 7^{10}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2Cs, 3B	$1$	$2^{4}$	$1$	$5.955778371$	2.105685636	$\frac{128787625}{98}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-43 a - 96$ , $-270 a - 307\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-43a-96\right){x}-270a-307$
686.1-d9	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{3} \cdot 7^{21}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3Cs	$1$	$2^{2} \cdot 3$	$1$	$1.985259457$	2.105685636	$\frac{392127492092318125}{55365148804} a + \frac{138814532776321000}{13841287201}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-2593 a - 4241$ , $94830 a + 138943\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2593a-4241\right){x}+94830a+138943$
686.1-d10	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{18} \cdot 7^{10}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2Cs, 3B	$1$	$2^{4} \cdot 3^{2}$	$1$	$0.661753152$	2.105685636	$\frac{2251439055699625}{25088}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-10923 a - 24576$ , $1213206 a + 1378643\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-10923a-24576\right){x}+1213206a+1378643$
686.1-d11	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2 \cdot 7^{11}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$4$	$2^{2}$	$1$	$1.488944592$	2.105685636	$\frac{2928743223192875}{4802} a + \frac{2070934198465000}{2401}$	$\bigl[1$ , $-a + 1$ , $1$ , $485 a - 759$ , $-2705 a + 3591\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(485a-759\right){x}-2705a+3591$
686.1-d12	686.1-d	$12$	$36$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{9} \cdot 7^{11}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	$2^{2} \cdot 3^{2}$	$1$	$0.661753152$	2.105685636	$\frac{218768831290078842857759125}{76832} a + \frac{9668307757341907702465000}{2401}$	$\bigl[1$ , $-a + 1$ , $1$ , $103060 a - 164599$ , $23909933 a - 32816619\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(103060a-164599\right){x}+23909933a-32816619$
686.1-e1	686.1-e	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{2} \cdot 7^{11}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2 \cdot 3^{2}$	$1$	$0.348888885$	2.220315274	$-\frac{341511377481251}{686} a - \frac{482970021761709}{686}$	$\bigl[a + 1$ , $a + 1$ , $a + 1$ , $-476 a - 696$ , $-7429 a - 10807\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-476a-696\right){x}-7429a-10807$
686.1-e2	686.1-e	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{6} \cdot 7^{9}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2 \cdot 3^{2}$	$1$	$3.139999973$	2.220315274	$-\frac{208183}{56} a - \frac{292563}{56}$	$\bigl[a + 1$ , $a + 1$ , $a + 1$ , $-6 a - 6$ , $-21 a - 19\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a-6\right){x}-21a-19$
686.1-f1	686.1-f	$2$	$2$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{10} \cdot 7^{4}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$1$	$3.016296194$	1.066421746	$\frac{2968038}{7} a - \frac{133931057}{224}$	$\bigl[a + 1$ , $1$ , $0$ , $9 a - 7$ , $16 a - 19\bigr]$	${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(9a-7\right){x}+16a-19$
686.1-f2	686.1-f	$2$	$2$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{5} \cdot 7^{5}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$1$	$3.016296194$	1.066421746	$-\frac{199003633361573}{392} a + \frac{35179204742368}{49}$	$\bigl[1$ , $-a + 1$ , $0$ , $-176 a - 256$ , $36 a + 32\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-176a-256\right){x}+36a+32$
686.1-g1	686.1-g	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{2} \cdot 7^{13}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	$2$	$1$	$3.130511611$	2.213605989	$\frac{7974621}{686} a - \frac{12367339}{686}$	$\bigl[1$ , $-a$ , $a$ , $13 a - 39$ , $-67 a + 51\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(13a-39\right){x}-67a+51$
686.1-g2	686.1-g	$2$	$3$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	686.1	$2 \cdot 7^{3}$	$2^{6} \cdot 7^{19}$	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	$2 \cdot 3$	$1$	$1.043503870$	2.213605989	$-\frac{59405903367}{322828856} a + \frac{391232728667}{322828856}$	$\bigl[1$ , $-a$ , $a$ , $-37 a + 216$ , $-64 a + 756\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-37a+216\right){x}-64a+756$

  Download displayed columns for results to

  *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.