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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
27.1-a2 27.1-a 6.6.1387029.1 \( 3^{3} \) $1$ $\Z/3\Z$ $1.089220952$ \( 3426222102 a^{5} - 12812570435 a^{4} + 2623723832 a^{3} + 28894116691 a^{2} - 24794656162 a + 4632631517 \) \( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( a^{5} - 3 a^{4} - 3 a^{3} + 10 a^{2} + 3 a - 3\) , \( -a^{5} + 3 a^{4} + 3 a^{3} - 9 a^{2} - 2 a + 3\) , \( -3 a^{5} - a^{4} + 10 a^{3} + 4 a^{2} + a - 1\) , \( -41 a^{5} + 17 a^{4} + 129 a^{3} - 31 a^{2} - 56 a + 14\bigr] \) ${y}^2+\left(a^{4}-2a^{3}-2a^{2}+4a\right){x}{y}+\left(-a^{5}+3a^{4}+3a^{3}-9a^{2}-2a+3\right){y}={x}^{3}+\left(a^{5}-3a^{4}-3a^{3}+10a^{2}+3a-3\right){x}^{2}+\left(-3a^{5}-a^{4}+10a^{3}+4a^{2}+a-1\right){x}-41a^{5}+17a^{4}+129a^{3}-31a^{2}-56a+14$
27.1-p2 27.1-p 6.6.1387029.1 \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( 3426222102 a^{5} - 12812570435 a^{4} + 2623723832 a^{3} + 28894116691 a^{2} - 24794656162 a + 4632631517 \) \( \bigl[a^{3} - 4 a + 1\) , \( 3 a^{5} - 7 a^{4} - 9 a^{3} + 18 a^{2} + 4 a - 5\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 7 a^{2} + 2\) , \( 7 a^{5} - 34 a^{4} + 40 a^{3} + 16 a^{2} - 31 a + 10\) , \( 53 a^{5} - 232 a^{4} + 239 a^{3} + 81 a^{2} - 149 a + 29\bigr] \) ${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(-a^{5}+3a^{4}+2a^{3}-7a^{2}+2\right){y}={x}^{3}+\left(3a^{5}-7a^{4}-9a^{3}+18a^{2}+4a-5\right){x}^{2}+\left(7a^{5}-34a^{4}+40a^{3}+16a^{2}-31a+10\right){x}+53a^{5}-232a^{4}+239a^{3}+81a^{2}-149a+29$
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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.