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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$\ \ell$
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Results (6 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
49.2-a1 49.2-a 4.4.4205.1 \( 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $102.9191920$ 1.587133170 \( \frac{91176666325}{282475249} a^{3} - \frac{91176666325}{282475249} a^{2} - \frac{547059997950}{282475249} a + \frac{199910878122}{282475249} \) \( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( 0\) , \( 0\) , \( -4 a^{3} + 4 a^{2} + 24 a - 8\) , \( -a^{3} + a^{2} + 6 a - 1\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}={x}^{3}+\left(-4a^{3}+4a^{2}+24a-8\right){x}-a^{3}+a^{2}+6a-1$
49.2-a2 49.2-a 4.4.4205.1 \( 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $411.6767683$ 1.587133170 \( -\frac{173650213}{16807} a^{3} + \frac{173650213}{16807} a^{2} + \frac{1041901278}{16807} a + \frac{583264453}{16807} \) \( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( 0\) , \( 0\) , \( a^{3} - a^{2} - 6 a + 2\) , \( 0\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}$
49.2-a3 49.2-a 4.4.4205.1 \( 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $102.9191920$ 1.587133170 \( \frac{213433415640625}{49} a^{3} - \frac{213433415640625}{49} a^{2} - \frac{1280600493843750}{49} a + \frac{467970351097797}{49} \) \( \bigl[1\) , \( a^{3} - a^{2} - 6 a - 2\) , \( 0\) , \( -27\) , \( 15 a^{3} - 15 a^{2} - 90 a - 5\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a^{3}-a^{2}-6a-2\right){x}^{2}-27{x}+15a^{3}-15a^{2}-90a-5$
49.2-a4 49.2-a 4.4.4205.1 \( 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $411.6767683$ 1.587133170 \( -\frac{94831363}{7} a^{3} + \frac{94831363}{7} a^{2} + \frac{568988178}{7} a + \frac{268859728}{7} \) \( \bigl[1\) , \( a^{3} - a^{2} - 6 a - 2\) , \( 0\) , \( 5 a^{3} - 5 a^{2} - 30 a - 17\) , \( 6 a^{3} - 6 a^{2} - 36 a - 19\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a^{3}-a^{2}-6a-2\right){x}^{2}+\left(5a^{3}-5a^{2}-30a-17\right){x}+6a^{3}-6a^{2}-36a-19$
49.2-b1 49.2-b 4.4.4205.1 \( 7^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.003877320$ $781.6042341$ 2.243245700 \( \frac{782646012}{117649} a^{3} - \frac{736229826}{117649} a^{2} - \frac{3610034665}{117649} a - \frac{1733848910}{117649} \) \( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( a + 1\) , \( 4 a^{3} - 7 a^{2} - 11 a\) , \( -7 a^{3} + 17 a^{2} + 8 a - 2\bigr] \) ${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+2\right){x}^{2}+\left(4a^{3}-7a^{2}-11a\right){x}-7a^{3}+17a^{2}+8a-2$
49.2-c1 49.2-c 4.4.4205.1 \( 7^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.003877320$ $781.6042341$ 2.243245700 \( -\frac{349611581}{117649} a^{3} + \frac{303195395}{117649} a^{2} + \frac{1011828079}{117649} a - \frac{369510387}{117649} \) \( \bigl[1\) , \( a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a\) , \( -33 a^{3} + 54 a^{2} + 131 a - 50\) , \( 134 a^{3} - 221 a^{2} - 526 a + 206\bigr] \) ${y}^2+{x}{y}+\left(-a^{3}+2a^{2}+3a\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a-1\right){x}^{2}+\left(-33a^{3}+54a^{2}+131a-50\right){x}+134a^{3}-221a^{2}-526a+206$
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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.