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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
13.1-a1 13.1-a 4.4.9909.1 \( 13 \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $1.079208123$ $441.3551652$ 2.126649835 \( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \) \( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -17 a^{3} - 3 a^{2} + 34 a - 30\) , \( 45 a^{3} + 106 a^{2} + 25 a - 23\bigr] \) ${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-17a^{3}-3a^{2}+34a-30\right){x}+45a^{3}+106a^{2}+25a-23$
13.1-a2 13.1-a 4.4.9909.1 \( 13 \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.359736041$ $441.3551652$ 2.126649835 \( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \) \( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -2 a^{3} + 2 a^{2} + 9 a - 5\) , \( -14 a^{3} + 17 a^{2} + 64 a - 35\bigr] \) ${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-2a^{3}+2a^{2}+9a-5\right){x}-14a^{3}+17a^{2}+64a-35$
13.1-a3 13.1-a 4.4.9909.1 \( 13 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1.079208123$ $5.448829201$ 2.126649835 \( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \) \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a\) , \( 0\) , \( 21 a^{3} + 8 a^{2} - 26 a - 9\) , \( 562 a^{3} + 1305 a^{2} + 277 a - 581\bigr] \) ${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+a{x}^{2}+\left(21a^{3}+8a^{2}-26a-9\right){x}+562a^{3}+1305a^{2}+277a-581$
13.1-b1 13.1-b 4.4.9909.1 \( 13 \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.277561838$ $403.6980252$ 1.500860066 \( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \) \( \bigl[a^{3} - 4 a - 2\) , \( a\) , \( a^{3} - 5 a - 1\) , \( 9 a^{3} - 18 a^{2} - 22 a + 14\) , \( -46 a^{3} + 74 a^{2} + 104 a - 66\bigr] \) ${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+a{x}^{2}+\left(9a^{3}-18a^{2}-22a+14\right){x}-46a^{3}+74a^{2}+104a-66$
13.1-b2 13.1-b 4.4.9909.1 \( 13 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.832685514$ $4.983926238$ 1.500860066 \( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \) \( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - a - 2\) , \( -130 a^{3} + 234 a^{2} + 353 a - 228\) , \( 477 a^{3} - 959 a^{2} - 995 a + 687\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-130a^{3}+234a^{2}+353a-228\right){x}+477a^{3}-959a^{2}-995a+687$
13.1-b3 13.1-b 4.4.9909.1 \( 13 \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.832685514$ $403.6980252$ 1.500860066 \( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \) \( \bigl[1\) , \( -a^{3} + a^{2} + 4 a\) , \( 1\) , \( -220 a^{3} + 274 a^{2} + 1002 a - 570\) , \( 2785 a^{3} - 3453 a^{2} - 12765 a + 7022\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-220a^{3}+274a^{2}+1002a-570\right){x}+2785a^{3}-3453a^{2}-12765a+7022$
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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.