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Results (24 matches)

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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
686.2-a1 686.2-a \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.136251981$ 2.008628703 \( -\frac{2968038}{7} a - \frac{133931057}{224} \) \( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -44 a - 6\) , \( -66 a - 135\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-44a-6\right){x}-66a-135$
686.2-a2 686.2-a \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.136251981$ 2.008628703 \( \frac{199003633361573}{392} a + \frac{35179204742368}{49} \) \( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -304 a - 646\) , \( -4930 a - 6831\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-304a-646\right){x}-4930a-6831$
686.2-b1 686.2-b \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.050461553$ $20.34834318$ 1.452127213 \( -\frac{7974621}{686} a - \frac{12367339}{686} \) \( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -6 a - 8\) , \( 6 a + 10\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6a-8\right){x}+6a+10$
686.2-b2 686.2-b \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.016820517$ $2.260927020$ 1.452127213 \( \frac{59405903367}{322828856} a + \frac{391232728667}{322828856} \) \( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 24 a + 47\) , \( 66 a + 71\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(24a+47\right){x}+66a+71$
686.2-c1 686.2-c \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.125216197$ $8.219192181$ 1.455474647 \( \frac{341511377481251}{686} a - \frac{482970021761709}{686} \) \( \bigl[1\) , \( -a\) , \( a + 1\) , \( 30 a - 50\) , \( -122 a + 124\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(30a-50\right){x}-122a+124$
686.2-c2 686.2-c \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.041738732$ $24.65757654$ 1.455474647 \( \frac{208183}{56} a - \frac{292563}{56} \) \( \bigl[1\) , \( -a\) , \( a + 1\) , \( 0\) , \( 0\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}$
686.2-d1 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $0.661753152$ 2.105685636 \( -\frac{548347731625}{1835008} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 681 a - 1536\) , \( -19223 a + 21843\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(681a-1536\right){x}-19223a+21843$
686.2-d2 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $5.955778371$ 2.105685636 \( -\frac{15625}{28} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( a - 6\) , \( 5 a - 7\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(a-6\right){x}+5a-7$
686.2-d3 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $1.985259457$ 2.105685636 \( \frac{9938375}{21952} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -19 a + 39\) , \( -127 a + 143\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-19a+39\right){x}-127a+143$
686.2-d4 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.488944592$ 2.105685636 \( -\frac{2928743223192875}{4802} a + \frac{2070934198465000}{2401} \) \( \bigl[1\) , \( a + 1\) , \( 1\) , \( -485 a - 759\) , \( 2705 a + 3591\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-485a-759\right){x}+2705a+3591$
686.2-d5 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.985259457$ 2.105685636 \( \frac{4956477625}{941192} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 141 a - 321\) , \( -1535 a + 1743\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(141a-321\right){x}-1535a+1743$
686.2-d6 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.985259457$ 2.105685636 \( -\frac{392127492092318125}{55365148804} a + \frac{138814532776321000}{13841287201} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 2591 a - 4241\) , \( -94831 a + 138943\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(2591a-4241\right){x}-94831a+138943$
686.2-d7 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.661753152$ 2.105685636 \( -\frac{218768831290078842857759125}{76832} a + \frac{9668307757341907702465000}{2401} \) \( \bigl[1\) , \( a + 1\) , \( 1\) , \( -103060 a - 164599\) , \( -23909933 a - 32816619\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-103060a-164599\right){x}-23909933a-32816619$
686.2-d8 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.955778371$ 2.105685636 \( \frac{128787625}{98} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 41 a - 96\) , \( 269 a - 307\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(41a-96\right){x}+269a-307$
686.2-d9 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.496314864$ 2.105685636 \( \frac{392127492092318125}{55365148804} a + \frac{138814532776321000}{13841287201} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 251 a - 2161\) , \( 5169 a - 37057\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(251a-2161\right){x}+5169a-37057$
686.2-d10 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.661753152$ 2.105685636 \( \frac{2251439055699625}{25088} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 10921 a - 24576\) , \( -1213207 a + 1378643\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(10921a-24576\right){x}-1213207a+1378643$
686.2-d11 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.955778371$ 2.105685636 \( \frac{2928743223192875}{4802} a + \frac{2070934198465000}{2401} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -134 a - 376\) , \( 1907 a + 2353\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-134a-376\right){x}+1907a+2353$
686.2-d12 686.2-d \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.165438288$ 2.105685636 \( \frac{218768831290078842857759125}{76832} a + \frac{9668307757341907702465000}{2401} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -37479 a - 93056\) , \( -8696055 a - 9208877\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-37479a-93056\right){x}-8696055a-9208877$
686.2-e1 686.2-e \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $0.348888885$ 2.220315274 \( \frac{341511377481251}{686} a - \frac{482970021761709}{686} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 478 a - 695\) , \( 6733 a - 9853\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(478a-695\right){x}+6733a-9853$
686.2-e2 686.2-e \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/3\Z$ $\mathrm{SU}(2)$ $1$ $3.139999973$ 2.220315274 \( \frac{208183}{56} a - \frac{292563}{56} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 8 a - 5\) , \( 15 a - 5\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a-5\right){x}+15a-5$
686.2-f1 686.2-f \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $3.016296194$ 1.066421746 \( -\frac{2968038}{7} a - \frac{133931057}{224} \) \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -9 a - 7\) , \( -16 a - 19\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9a-7\right){x}-16a-19$
686.2-f2 686.2-f \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $3.016296194$ 1.066421746 \( \frac{199003633361573}{392} a + \frac{35179204742368}{49} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( 176 a - 256\) , \( -36 a + 32\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(176a-256\right){x}-36a+32$
686.2-g1 686.2-g \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $3.130511611$ 2.213605989 \( -\frac{7974621}{686} a - \frac{12367339}{686} \) \( \bigl[1\) , \( a\) , \( a\) , \( -14 a - 39\) , \( 67 a + 51\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-14a-39\right){x}+67a+51$
686.2-g2 686.2-g \(\Q(\sqrt{2}) \) \( 2 \cdot 7^{3} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $1.043503870$ 2.213605989 \( \frac{59405903367}{322828856} a + \frac{391232728667}{322828856} \) \( \bigl[1\) , \( a\) , \( a\) , \( 36 a + 216\) , \( 64 a + 756\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(36a+216\right){x}+64a+756$
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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.