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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 (8 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
32.1-a1 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $-16$ $N(\mathrm{U}(1))$ $1$ $189.0727201$ 1.044489082 \( 287496 \) \( \bigl[a^{2} - 2\) , \( 1\) , \( 0\) , \( -2\) , \( -3\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+{x}^{2}-2{x}-3$
32.1-a2 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/2\Z\oplus\Z/8\Z$ $-16$ $N(\mathrm{U}(1))$ $1$ $3025.163522$ 1.044489082 \( 287496 \) \( \bigl[a^{2} - 2\) , \( 1\) , \( a^{2} - 2\) , \( -3\) , \( 0\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+{x}^{2}-3{x}$
32.1-a3 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/8\Z$ $-64$ $N(\mathrm{U}(1))$ $1$ $1512.581761$ 1.044489082 \( 29071392966 a^{2} - 17029627812 \) \( \bigl[a^{3} - 3 a\) , \( a^{2} - 1\) , \( a^{3} - 2 a\) , \( a^{2} - 6\) , \( -6 a^{2} + 19\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(a^{2}-6\right){x}-6a^{2}+19$
32.1-a4 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/2\Z$ $-64$ $N(\mathrm{U}(1))$ $1$ $23.63409001$ 1.044489082 \( -29071392966 a^{2} + 99255944052 \) \( \bigl[a^{3} - 3 a\) , \( a^{2} - 1\) , \( a\) , \( -3\) , \( -6 a^{2}\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-1\right){x}^{2}-3{x}-6a^{2}$
32.1-a5 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/4\Z$ $-4$ $N(\mathrm{U}(1))$ $1$ $378.1454402$ 1.044489082 \( 1728 \) \( \bigl[a^{3} - 2 a\) , \( a^{2} - 3\) , \( 0\) , \( -a^{2} + 5\) , \( a^{2} - 3\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-a^{2}+5\right){x}+a^{2}-3$
32.1-a6 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/2\Z\oplus\Z/4\Z$ $-4$ $N(\mathrm{U}(1))$ $1$ $1512.581761$ 1.044489082 \( 1728 \) \( \bigl[a^{3} - 2 a\) , \( a^{2} - 3\) , \( a^{3} - 2 a\) , \( -2\) , \( -a^{2} + 2\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}-2{x}-a^{2}+2$
32.1-a7 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/2\Z$ $-64$ $N(\mathrm{U}(1))$ $1$ $23.63409001$ 1.044489082 \( 29071392966 a^{2} - 17029627812 \) \( \bigl[a\) , \( -a^{2} + 3\) , \( a^{3} - 3 a\) , \( -a^{2} - 1\) , \( 6 a^{2} - 24\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{2}-1\right){x}+6a^{2}-24$
32.1-a8 32.1-a \(\Q(\zeta_{16})^+\) \( 2^{5} \) 0 $\Z/8\Z$ $-64$ $N(\mathrm{U}(1))$ $1$ $1512.581761$ 1.044489082 \( -29071392966 a^{2} + 99255944052 \) \( \bigl[a\) , \( -a^{2} + 3\) , \( a^{3} - 2 a\) , \( -2 a^{2}\) , \( 5 a^{2} - 3\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}-2a^{2}{x}+5a^{2}-3$
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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.