↑

Learn more

Refine search

Base field Conductor norm Rank* Torsion order CM discriminant
Base field degree/signature Bad \(p\) $\Q$-curves Torsion structure CM
Analytic order of Ш* Cyclic isogeny degree Isogeny class size Reduction Galois image
j-invariant Regulator* Curves per isogeny class Isogeny class degree Nonmax$\ \ell$
Sort order Select

Results (10 matches)

  displayed columns for results
Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
961.7-a1 961.7-a \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $0.084970583$ $257.0130160$ 2.604398559 \( 0 \) \( \bigl[0\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -2 a^{3} - a^{2} + 7 a + 3\) , \( -7541548 a^{3} - 6237551 a^{2} + 18769617 a + 4127623\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(-2a^{3}-a^{2}+7a+3\right){x}-7541548a^{3}-6237551a^{2}+18769617a+4127623$
961.7-a2 961.7-a \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $0.028323527$ $771.0390481$ 2.604398559 \( 0 \) \( \bigl[0\) , \( a^{3} - 2 a + 1\) , \( a^{3} - 3 a + 1\) , \( a^{3} + a^{2} - 3 a\) , \( a^{3} + a^{2} - 2 a - 1\bigr] \) ${y}^2+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}+a^{3}+a^{2}-2a-1$
961.7-b1 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $0.111217262$ $388.2543833$ 2.574792903 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{2} + a - 2\) , \( -742 a^{3} + 466 a^{2} + 2513 a - 2417\) , \( 13104 a^{3} - 11905 a^{2} - 46670 a + 52371\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-742a^{3}+466a^{2}+2513a-2417\right){x}+13104a^{3}-11905a^{2}-46670a+52371$
961.7-b2 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $0.055608631$ $388.2543833$ 2.574792903 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{2} + a - 2\) , \( -47 a^{3} + 31 a^{2} + 158 a - 152\) , \( 196 a^{3} - 209 a^{2} - 719 a + 867\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-47a^{3}+31a^{2}+158a-152\right){x}+196a^{3}-209a^{2}-719a+867$
961.7-b3 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $0.556086312$ $77.65087667$ 2.574792903 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{2} - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -2 a^{3} - 17 a^{2} + 53 a - 39\) , \( 28 a^{3} - 155 a^{2} + 201 a - 55\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(-2a^{3}-17a^{2}+53a-39\right){x}+28a^{3}-155a^{2}+201a-55$
961.7-b4 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $0.278043156$ $77.65087667$ 2.574792903 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{2} - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 3 a^{3} - 2 a^{2} - 7 a + 6\) , \( -a^{2} + 8 a - 9\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(3a^{3}-2a^{2}-7a+6\right){x}-a^{2}+8a-9$
961.7-b5 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $0.166825893$ $129.4181277$ 2.574792903 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{3} - 3 a + 1\) , \( -a^{2} + 3\) , \( 1\) , \( 140176 a^{3} + 115937 a^{2} - 348875 a - 76717\) , \( 51582345 a^{3} + 42663288 a^{2} - 128379673 a - 28231951\bigr] \) ${y}^2+\left(a^{3}-3a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(140176a^{3}+115937a^{2}-348875a-76717\right){x}+51582345a^{3}+42663288a^{2}-128379673a-28231951$
961.7-b6 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $0.333651787$ $129.4181277$ 2.574792903 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{3} - 3 a + 1\) , \( -a^{2} + 3\) , \( 1\) , \( -945799 a^{3} - 782263 a^{2} + 2353930 a + 517653\) , \( 524197265 a^{3} + 433558794 a^{2} - 1304637727 a - 286902673\bigr] \) ${y}^2+\left(a^{3}-3a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-945799a^{3}-782263a^{2}+2353930a+517653\right){x}+524197265a^{3}+433558794a^{2}-1304637727a-286902673$
961.7-b7 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $0.834129469$ $25.88362555$ 2.574792903 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 1\) , \( -72303 a^{3} - 59800 a^{2} + 179950 a + 39572\) , \( -14619624 a^{3} - 12091758 a^{2} + 36385754 a + 8001585\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}+a^{2}-4a-1\right){x}^{2}+\left(-72303a^{3}-59800a^{2}+179950a+39572\right){x}-14619624a^{3}-12091758a^{2}+36385754a+8001585$
961.7-b8 961.7-b \(\Q(\zeta_{15})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1.668258938$ $25.88362555$ 2.574792903 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 1\) , \( -1158278 a^{3} - 958000 a^{2} + 2882755 a + 633942\) , \( -931622486 a^{3} - 770536492 a^{2} + 2318649718 a + 509893883\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}+a^{2}-4a-1\right){x}^{2}+\left(-1158278a^{3}-958000a^{2}+2882755a+633942\right){x}-931622486a^{3}-770536492a^{2}+2318649718a+509893883$
  displayed columns for results

  *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.