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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-a1 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $226.2402485$ 0.843147624 \( \frac{13262138608154212980818020352}{923521} a^{3} - \frac{16034680544860826249617094257}{923521} a^{2} - \frac{49696393399932989894285277456}{923521} a + \frac{63437929696698697519404008684}{923521} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -427 a^{3} - 156 a^{2} + 1210 a - 328\) , \( 5301 a^{3} + 2573 a^{2} - 14446 a + 2048\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-427a^{3}-156a^{2}+1210a-328\right){x}+5301a^{3}+2573a^{2}-14446a+2048$
31.1-b1 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2.959428063$ 0.705864781 \( \frac{13262138608154212980818020352}{923521} a^{3} - \frac{16034680544860826249617094257}{923521} a^{2} - \frac{49696393399932989894285277456}{923521} a + \frac{63437929696698697519404008684}{923521} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -80 a^{3} + 335 a^{2} + 315 a - 1284\) , \( -2177 a^{3} + 4753 a^{2} + 8378 a - 18117\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-80a^{3}+335a^{2}+315a-1284\right){x}-2177a^{3}+4753a^{2}+8378a-18117$
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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.