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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
83.1-a1 83.1-a 6.6.592661.1 \( 83 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.005387149$ $80327.79265$ 3.37267 \( \frac{2284010860}{83} a^{5} + \frac{884600390}{83} a^{4} - \frac{10195196217}{83} a^{3} - \frac{5005268270}{83} a^{2} + \frac{4484649386}{83} a + \frac{1645381845}{83} \) \( \bigl[a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( -a^{4} - a^{3} + 5 a^{2} + 3 a - 2\) , \( a^{2} - 2\) , \( -a^{5} + 2 a^{4} + 6 a^{3} + a - 2\) , \( a^{5} + 7 a^{2} - 2 a - 2\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{4}-a^{3}+5a^{2}+3a-2\right){x}^{2}+\left(-a^{5}+2a^{4}+6a^{3}+a-2\right){x}+a^{5}+7a^{2}-2a-2$
83.1-b1 83.1-b 6.6.592661.1 \( 83 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.026258286$ $14031.16581$ 2.87149 \( -\frac{166519119719}{83} a^{5} + \frac{57331816854}{83} a^{4} + \frac{870188329038}{83} a^{3} - \frac{95490374048}{83} a^{2} - \frac{895208628138}{83} a - \frac{253954106815}{83} \) \( \bigl[a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 4 a - 1\) , \( -a^{4} + a^{3} + 5 a^{2} - 4 a - 4\) , \( 2 a^{5} - a^{4} - 9 a^{3} + 4 a^{2} + 5 a - 2\) , \( 3 a^{5} - 14 a^{3} - a^{2} + 10 a + 1\) , \( -3 a^{5} - a^{4} + 13 a^{3} + 6 a^{2} - 5 a - 4\bigr] \) ${y}^2+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+4a-1\right){x}{y}+\left(2a^{5}-a^{4}-9a^{3}+4a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-4a-4\right){x}^{2}+\left(3a^{5}-14a^{3}-a^{2}+10a+1\right){x}-3a^{5}-a^{4}+13a^{3}+6a^{2}-5a-4$
83.1-c1 83.1-c 6.6.592661.1 \( 83 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.007707327$ $12401.08843$ 2.97969 \( \frac{15736963093640}{47458321} a^{5} - \frac{44917566551496}{47458321} a^{4} + \frac{6514884455608}{47458321} a^{3} + \frac{47332089860599}{47458321} a^{2} - \frac{11320884612420}{47458321} a - \frac{8429605989992}{47458321} \) \( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 4\) , \( -a^{5} + 2 a^{4} + 4 a^{3} - 8 a^{2} - a + 2\) , \( 2 a^{5} - a^{4} - 9 a^{3} + 5 a^{2} + 6 a - 4\) , \( 2 a^{5} + 5 a^{4} - 10 a^{3} - 23 a^{2} + 6 a + 7\) , \( -21 a^{5} + 13 a^{4} + 84 a^{3} - 42 a^{2} - 17 a + 6\bigr] \) ${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-4\right){x}{y}+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+6a-4\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+4a^{3}-8a^{2}-a+2\right){x}^{2}+\left(2a^{5}+5a^{4}-10a^{3}-23a^{2}+6a+7\right){x}-21a^{5}+13a^{4}+84a^{3}-42a^{2}-17a+6$
83.1-d1 83.1-d 6.6.592661.1 \( 83 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.005613072$ $70481.43116$ 3.08335 \( \frac{46638196}{83} a^{5} + \frac{119264171}{83} a^{4} - \frac{90683712}{83} a^{3} - \frac{273899140}{83} a^{2} + \frac{36697985}{83} a + \frac{37204710}{83} \) \( \bigl[2 a^{5} - 2 a^{4} - 9 a^{3} + 9 a^{2} + 5 a - 6\) , \( a^{5} - 5 a^{3} + a^{2} + 3 a - 4\) , \( a^{4} - 4 a^{2} + a + 1\) , \( -11 a^{5} + 14 a^{4} + 48 a^{3} - 56 a^{2} - 32 a + 32\) , \( -23 a^{5} + 31 a^{4} + 105 a^{3} - 128 a^{2} - 76 a + 72\bigr] \) ${y}^2+\left(2a^{5}-2a^{4}-9a^{3}+9a^{2}+5a-6\right){x}{y}+\left(a^{4}-4a^{2}+a+1\right){y}={x}^{3}+\left(a^{5}-5a^{3}+a^{2}+3a-4\right){x}^{2}+\left(-11a^{5}+14a^{4}+48a^{3}-56a^{2}-32a+32\right){x}-23a^{5}+31a^{4}+105a^{3}-128a^{2}-76a+72$
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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.