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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
686.1-a1 686.1-a \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $1.136251981$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $1.136251981$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\Z/3\Z$ $0.050461553$ $20.34834318$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\mathsf{trivial}$ $0.016820517$ $2.260927020$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\mathsf{trivial}$ $0.125216197$ $8.219192181$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\mathsf{trivial}$ $0.041738732$ $24.65757654$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/4\Z$ $1$ $0.661753152$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/4\Z$ $1$ $5.955778371$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/4\Z$ $1$ $1.985259457$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $5.955778371$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $1.985259457$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $0.496314864$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $0.165438288$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $5.955778371$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $1.985259457$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $0.661753152$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $1.488944592$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $0.661753152$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\mathsf{trivial}$ $1$ $0.348888885$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/3\Z$ $1$ $3.139999973$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $3.016296194$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\Z/2\Z$ $1$ $3.016296194$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\mathsf{trivial}$ $1$ $3.130511611$ 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 \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $0$ $\mathsf{trivial}$ $1$ $1.043503870$ 2.213605989 \( -\frac{59405903367}{322828856} a + \frac{391232728667}{322828856} \) \( \bigl[1\) , \( -a\) , \( a\) , \( -37 a + 216\) , \( -64 a + 756\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-37a+216\right){x}-64a+756$
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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.