↑

Learn more

Refine search

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$\ \ell$
Sort order Select

Results (12 matches)

  displayed columns for results
Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
1.1-a6 1.1-a \(\Q(\zeta_{16})^+\) \( 1 \) 0 $\Z/10\Z$ $\mathrm{SU}(2)$ $1$ $1018.917972$ 0.225151189 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{2} + a - 2\) , \( a^{2} - a - 1\) , \( 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -9 a^{3} + 17 a^{2} + 5 a - 10\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(a^{3}-a^{2}-2a+2\right){x}-9a^{3}+17a^{2}+5a-10$
16.1-a3 16.1-a \(\Q(\zeta_{16})^+\) \( 2^{4} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $154.9911952$ 0.856213478 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 4 a^{3} - 2 a^{2} - 14 a + 3\) , \( 4 a^{3} - 6 a^{2} - 9 a + 9\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(4a^{3}-2a^{2}-14a+3\right){x}+4a^{3}-6a^{2}-9a+9$
256.1-b7 256.1-b \(\Q(\zeta_{16})^+\) \( 2^{8} \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $1124.004677$ 1.552326815 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 3 a + 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 2 a^{3} - 4 a^{2} - 2 a + 3\) , \( -30 a^{3} + 55 a^{2} + 17 a - 33\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+1\right){x}^{2}+\left(2a^{3}-4a^{2}-2a+3\right){x}-30a^{3}+55a^{2}+17a-33$
256.1-d6 256.1-d \(\Q(\zeta_{16})^+\) \( 2^{8} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $70.25029234$ 1.552326815 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{2} - 2\) , \( a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 2 a^{3} - 4 a^{2} - 2 a + 3\) , \( 30 a^{3} - 56 a^{2} - 19 a + 32\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-2\right){x}^{2}+\left(2a^{3}-4a^{2}-2a+3\right){x}+30a^{3}-56a^{2}-19a+32$
289.10-a12 289.10-a \(\Q(\zeta_{16})^+\) \( 17^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $68.15279425$ 3.011956435 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a\) , \( a^{2} - 2\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 4 a^{3} - a^{2} - 12 a - 5\) , \( -11 a^{3} + 26 a^{2} - 3 a - 23\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(4a^{3}-a^{2}-12a-5\right){x}-11a^{3}+26a^{2}-3a-23$
289.7-a4 289.7-a \(\Q(\zeta_{16})^+\) \( 17^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $272.6111770$ 3.011956435 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} - a^{2} + 3 a + 3\) , \( a^{2} - 1\) , \( 44 a^{3} + 32 a^{2} - 151 a - 112\) , \( -153 a^{3} - 118 a^{2} + 523 a + 401\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+3\right){x}^{2}+\left(44a^{3}+32a^{2}-151a-112\right){x}-153a^{3}-118a^{2}+523a+401$
289.8-a6 289.8-a \(\Q(\zeta_{16})^+\) \( 17^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $136.3055885$ 3.011956435 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{2} + a - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{3} - 2 a + 1\) , \( 12 a^{3} - 10 a^{2} - 41 a + 33\) , \( -6 a^{3} - 2 a^{2} + 12 a - 5\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-2a-2\right){x}^{2}+\left(12a^{3}-10a^{2}-41a+33\right){x}-6a^{3}-2a^{2}+12a-5$
289.9-a12 289.9-a \(\Q(\zeta_{16})^+\) \( 17^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $136.3055885$ 3.011956435 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a\) , \( -a^{3} + 3 a - 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -9 a^{2} - 3 a + 8\) , \( 26 a^{3} - 32 a^{2} - 14 a + 17\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-9a^{2}-3a+8\right){x}+26a^{3}-32a^{2}-14a+17$
961.10-b9 961.10-b \(\Q(\zeta_{16})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.381094116$ $106.0160756$ 1.785537548 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{3} - 3 a\) , \( -a^{2} - a + 2\) , \( a + 1\) , \( 6 a^{3} - 5 a^{2} - 29 a + 1\) , \( -7 a^{3} - 17 a^{2} + 12 a + 36\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(6a^{3}-5a^{2}-29a+1\right){x}-7a^{3}-17a^{2}+12a+36$
961.7-b11 961.7-b \(\Q(\zeta_{16})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.095273529$ $212.0321512$ 1.785537548 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a\) , \( a\) , \( a^{3} - 2 a + 1\) , \( 12 a^{3} + 5 a^{2} - 42 a - 21\) , \( -24 a^{3} - 19 a^{2} + 82 a + 62\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+a{x}^{2}+\left(12a^{3}+5a^{2}-42a-21\right){x}-24a^{3}-19a^{2}+82a+62$
961.8-b7 961.8-b \(\Q(\zeta_{16})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.762188233$ $26.50401890$ 1.785537548 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{2} + a - 2\) , \( -a^{3} + 4 a\) , \( a^{3} - 2 a + 1\) , \( -6 a^{3} - 16 a^{2} + 13\) , \( -33 a^{3} - 68 a^{2} + 20 a + 38\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-6a^{3}-16a^{2}+13\right){x}-33a^{3}-68a^{2}+20a+38$
961.9-b11 961.9-b \(\Q(\zeta_{16})^+\) \( 31^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.762188233$ $53.00803780$ 1.785537548 \( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) \( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 1\) , \( 1\) , \( 36 a^{3} - 30 a^{2} - 123 a + 100\) , \( -32 a^{3} + 21 a^{2} + 116 a - 86\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(36a^{3}-30a^{2}-123a+100\right){x}-32a^{3}+21a^{2}+116a-86$
  displayed columns for results

  *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.