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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
7.1-a1 7.1-a 4.4.13888.1 \( 7 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $223.0004007$ 1.892281700 \( \frac{37729567370}{21} a^{3} + \frac{49648197854}{21} a^{2} - \frac{99317907635}{21} a - \frac{34163722112}{7} \) \( \bigl[a\) , \( -a^{2} + 5\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{4}{3} a - 3\) , \( -\frac{5}{3} a^{3} + \frac{7}{3} a^{2} + \frac{20}{3} a - 4\) , \( -\frac{14}{3} a^{3} + \frac{37}{3} a^{2} + \frac{50}{3} a - 39\bigr] \) ${y}^2+a{x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{4}{3}a-3\right){y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-\frac{5}{3}a^{3}+\frac{7}{3}a^{2}+\frac{20}{3}a-4\right){x}-\frac{14}{3}a^{3}+\frac{37}{3}a^{2}+\frac{50}{3}a-39$
7.1-a2 7.1-a 4.4.13888.1 \( 7 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $223.0004007$ 1.892281700 \( -\frac{1081509563885}{1029} a^{3} + \frac{3140764344406}{1029} a^{2} + \frac{4731169352873}{1029} a - \frac{3588809847660}{343} \) \( \bigl[\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{1}{3} a + 1\) , \( \frac{1}{3} a^{3} - \frac{5}{3} a^{2} - \frac{4}{3} a + 5\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{7}{3} a - 2\) , \( -12 a^{3} + 35 a^{2} + 49 a - 124\) , \( -\frac{266}{3} a^{3} + \frac{688}{3} a^{2} + \frac{1274}{3} a - 704\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}-\frac{2}{3}a^{2}-\frac{1}{3}a+1\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{7}{3}a-2\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}-\frac{5}{3}a^{2}-\frac{4}{3}a+5\right){x}^{2}+\left(-12a^{3}+35a^{2}+49a-124\right){x}-\frac{266}{3}a^{3}+\frac{688}{3}a^{2}+\frac{1274}{3}a-704$
7.1-b1 7.1-b 4.4.13888.1 \( 7 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.012743831$ $2178.566117$ 2.827041244 \( -\frac{1081509563885}{1029} a^{3} + \frac{3140764344406}{1029} a^{2} + \frac{4731169352873}{1029} a - \frac{3588809847660}{343} \) \( \bigl[a\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{4}{3} a - 3\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{7}{3} a - 3\) , \( \frac{5}{3} a^{3} - \frac{7}{3} a^{2} - \frac{5}{3} a + 2\) , \( -\frac{16}{3} a^{3} + \frac{29}{3} a^{2} + \frac{100}{3} a + 18\bigr] \) ${y}^2+a{x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{7}{3}a-3\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{4}{3}a-3\right){x}^{2}+\left(\frac{5}{3}a^{3}-\frac{7}{3}a^{2}-\frac{5}{3}a+2\right){x}-\frac{16}{3}a^{3}+\frac{29}{3}a^{2}+\frac{100}{3}a+18$
7.1-b2 7.1-b 4.4.13888.1 \( 7 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.038231493$ $2178.566117$ 2.827041244 \( \frac{37729567370}{21} a^{3} + \frac{49648197854}{21} a^{2} - \frac{99317907635}{21} a - \frac{34163722112}{7} \) \( \bigl[\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{4}{3} a + 2\) , \( a^{2} - 2 a - 5\) , \( a\) , \( -\frac{4}{3} a^{3} - \frac{1}{3} a^{2} + \frac{13}{3} a + 2\) , \( \frac{2}{3} a^{3} + \frac{5}{3} a^{2} + \frac{10}{3} a + 5\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}-\frac{2}{3}a^{2}-\frac{4}{3}a+2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-2a-5\right){x}^{2}+\left(-\frac{4}{3}a^{3}-\frac{1}{3}a^{2}+\frac{13}{3}a+2\right){x}+\frac{2}{3}a^{3}+\frac{5}{3}a^{2}+\frac{10}{3}a+5$
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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.