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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 (10 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
841.9-a1 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{2} - 2\) , \( -290652 a^{3} - 240393 a^{2} + 723388 a + 159079\) , \( -118023233 a^{3} - 97615947 a^{2} + 293739721 a + 64596254\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-2a-2\right){x}^{2}+\left(-290652a^{3}-240393a^{2}+723388a+159079\right){x}-118023233a^{3}-97615947a^{2}+293739721a+64596254$
841.9-a2 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{2} - 2\) , \( -4656162 a^{3} - 3851083 a^{2} + 11588363 a + 2548394\) , \( -7511798086 a^{3} - 6212939993 a^{2} + 18695586113 a + 4111343335\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-2a-2\right){x}^{2}+\left(-4656162a^{3}-3851083a^{2}+11588363a+2548394\right){x}-7511798086a^{3}-6212939993a^{2}+18695586113a+4111343335$
841.9-a3 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 1\) , \( a^{3} - 3 a + 1\) , \( 3 a^{3} - 4 a^{2} - 24 a - 18\) , \( 9 a^{3} - 14 a^{2} - 72 a - 1\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(3a^{3}-4a^{2}-24a-18\right){x}+9a^{3}-14a^{2}-72a-1$
841.9-a4 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 1\) , \( a^{3} - 3 a + 1\) , \( 3 a^{3} + a^{2} - 9 a + 2\) , \( -3 a^{3} - 3 a^{2} + 7 a + 3\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(3a^{3}+a^{2}-9a+2\right){x}-3a^{3}-3a^{2}+7a+3$
841.9-a5 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{3} - 2 a\) , \( a^{3} + a^{2} - 4 a - 1\) , \( a^{2} + a - 2\) , \( 12100 a^{3} + 9958 a^{2} - 30230 a - 6644\) , \( 1300517 a^{3} + 1076015 a^{2} - 3235972 a - 711648\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-1\right){x}^{2}+\left(12100a^{3}+9958a^{2}-30230a-6644\right){x}+1300517a^{3}+1076015a^{2}-3235972a-711648$
841.9-a6 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{3} - 2 a\) , \( a^{3} + a^{2} - 4 a - 1\) , \( a^{2} + a - 2\) , \( -79220 a^{3} - 66357 a^{2} + 195350 a + 43016\) , \( 13008496 a^{3} + 10780866 a^{2} - 32329042 a - 7110931\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-1\right){x}^{2}+\left(-79220a^{3}-66357a^{2}+195350a+43016\right){x}+13008496a^{3}+10780866a^{2}-32329042a-7110931$
841.9-a7 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[a + 1\) , \( a^{2} - 3\) , \( a^{3} - 3 a\) , \( -4 a^{3} - 7 a^{2} - 2 a - 16\) , \( 10 a^{3} + 27 a^{2} + 14 a - 12\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-4a^{3}-7a^{2}-2a-16\right){x}+10a^{3}+27a^{2}+14a-12$
841.9-a8 841.9-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[a + 1\) , \( a^{2} - 3\) , \( a^{3} - 3 a\) , \( -4 a^{3} - 2 a^{2} + 13 a + 4\) , \( -3 a^{3} - a^{2} + 11 a + 2\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-4a^{3}-2a^{2}+13a+4\right){x}-3a^{3}-a^{2}+11a+2$
841.9-b1 841.9-b \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $1$ $47.68821090$ 1.421787750 \( 0 \) \( \bigl[0\) , \( -a^{2} - a + 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( a^{3} + a^{2} - 2 a\) , \( -16 a^{3} + 18 a^{2} + 49 a - 60\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}-16a^{3}+18a^{2}+49a-60$
841.9-b2 841.9-b \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $1$ $47.68821090$ 1.421787750 \( 0 \) \( \bigl[0\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( -a + 2\) , \( -815183 a^{3} - 674230 a^{2} + 2028854 a + 446165\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-a+2\right){x}-815183a^{3}-674230a^{2}+2028854a+446165$
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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.