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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
31.1-a5 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $226.2402485$ 0.843147624 \( -\frac{2933497938285}{923521} a^{3} + \frac{9133676668039}{923521} a^{2} - \frac{6067822613957}{923521} a - \frac{1204128338708}{923521} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -22 a^{3} - 21 a^{2} + 60 a + 12\) , \( -77 a^{3} - 67 a^{2} + 195 a + 43\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-22a^{3}-21a^{2}+60a+12\right){x}-77a^{3}-67a^{2}+195a+43$
31.1-b5 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z\oplus\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $757.6135842$ 0.705864781 \( -\frac{2933497938285}{923521} a^{3} + \frac{9133676668039}{923521} a^{2} - \frac{6067822613957}{923521} a - \frac{1204128338708}{923521} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -4\) , \( -a^{3} + 4 a^{2} + 4 a - 7\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}-4{x}-a^{3}+4a^{2}+4a-7$
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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.