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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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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
27.1-a1 27.1-a 5.5.65657.1 \( 3^{3} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $503.6602642$ 1.96560918 \( -169969542 a^{4} + 461869972 a^{3} + 56647563 a^{2} - 437225155 a - 98969253 \) \( \bigl[a^{3} - 4 a\) , \( 2 a^{4} - 3 a^{3} - 9 a^{2} + 10 a + 6\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 7\) , \( 2 a^{4} - a^{3} - 15 a^{2} + 10 a + 16\) , \( a^{3} - 4 a^{2} + 4 a + 3\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(-2a^{4}+3a^{3}+9a^{2}-8a-7\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-9a^{2}+10a+6\right){x}^{2}+\left(2a^{4}-a^{3}-15a^{2}+10a+16\right){x}+a^{3}-4a^{2}+4a+3$
27.1-f1 27.1-f 5.5.65657.1 \( 3^{3} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.006570846$ $5293.284853$ 2.03609173 \( -169969542 a^{4} + 461869972 a^{3} + 56647563 a^{2} - 437225155 a - 98969253 \) \( \bigl[a^{3} - 3 a - 1\) , \( -3 a^{4} + 4 a^{3} + 13 a^{2} - 10 a - 10\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 7\) , \( 2 a^{4} - 3 a^{3} - 12 a^{2} + 12 a + 19\) , \( -8 a^{4} + 9 a^{3} + 37 a^{2} - 21 a - 25\bigr] \) ${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(-2a^{4}+3a^{3}+9a^{2}-8a-7\right){y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-10a-10\right){x}^{2}+\left(2a^{4}-3a^{3}-12a^{2}+12a+19\right){x}-8a^{4}+9a^{3}+37a^{2}-21a-25$
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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.