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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 (4 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
25.3-a1 25.3-a 4.4.4525.1 \( 5^{2} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $108.7523764$ 1.616700096 \( -\frac{13556}{3} a^{3} + \frac{1736}{3} a^{2} + \frac{77018}{3} a + 12361 \) \( \bigl[\frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{3} a\) , \( a^{2} - 5\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a\) , \( -\frac{7}{3} a^{3} + \frac{19}{3} a^{2} + \frac{13}{3} a - 11\) , \( -\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{4}{3} a - 2\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{4}{3}a\right){x}{y}+\left(\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a\right){y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(-\frac{7}{3}a^{3}+\frac{19}{3}a^{2}+\frac{13}{3}a-11\right){x}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{4}{3}a-2$
25.3-a2 25.3-a 4.4.4525.1 \( 5^{2} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $108.7523764$ 1.616700096 \( -\frac{18442}{3} a^{3} - \frac{15974}{3} a^{2} + \frac{129295}{3} a + 47469 \) \( \bigl[\frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a\) , \( -\frac{1}{3} a^{3} - \frac{2}{3} a^{2} + \frac{10}{3} a + 3\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 3\) , \( 2 a^{3} + 3 a^{2} - a - 2\) , \( \frac{17}{3} a^{3} + \frac{28}{3} a^{2} - \frac{35}{3} a - 17\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-3\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{2}{3}a^{2}+\frac{10}{3}a+3\right){x}^{2}+\left(2a^{3}+3a^{2}-a-2\right){x}+\frac{17}{3}a^{3}+\frac{28}{3}a^{2}-\frac{35}{3}a-17$
25.3-b1 25.3-b 4.4.4525.1 \( 5^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.021477794$ $426.2423810$ 1.633120637 \( -\frac{18442}{3} a^{3} - \frac{15974}{3} a^{2} + \frac{129295}{3} a + 47469 \) \( \bigl[a^{2} - a - 4\) , \( -\frac{1}{3} a^{3} + \frac{4}{3} a^{2} + \frac{1}{3} a - 4\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a\) , \( 2 a^{2} - 4 a - 3\) , \( -\frac{2}{3} a^{3} + \frac{2}{3} a^{2} + \frac{8}{3} a - 1\bigr] \) ${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{4}{3}a^{2}+\frac{1}{3}a-4\right){x}^{2}+\left(2a^{2}-4a-3\right){x}-\frac{2}{3}a^{3}+\frac{2}{3}a^{2}+\frac{8}{3}a-1$
25.3-b2 25.3-b 4.4.4525.1 \( 5^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.064433383$ $426.2423810$ 1.633120637 \( -\frac{13556}{3} a^{3} + \frac{1736}{3} a^{2} + \frac{77018}{3} a + 12361 \) \( \bigl[\frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{3} a + 1\) , \( -a^{2} + 2 a + 4\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 4\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{13}{3} a - 3\) , \( \frac{10}{3} a^{3} + \frac{2}{3} a^{2} - \frac{64}{3} a - 21\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{4}{3}a+1\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-4\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{13}{3}a-3\right){x}+\frac{10}{3}a^{3}+\frac{2}{3}a^{2}-\frac{64}{3}a-21$
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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.