↑

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
64.1-a1 64.1-a \(\Q(\zeta_{20})^+\) \( 2^{6} \) 0 $\Z/8\Z$ $\mathrm{SU}(2)$ $1$ $1471.388299$ 1.028163831 \( -548896 a^{2} + 1987376 \) \( \bigl[a^{2} + a - 3\) , \( a^{2} - 4\) , \( a^{3} - 2 a + 1\) , \( 4 a^{3} + 5 a^{2} - 9 a - 9\) , \( 2 a^{2} + 7 a + 5\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(4a^{3}+5a^{2}-9a-9\right){x}+2a^{2}+7a+5$
64.1-a2 64.1-a \(\Q(\zeta_{20})^+\) \( 2^{6} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $22.99044218$ 1.028163831 \( 2711191688 a^{2} - 3746774764 \) \( \bigl[a^{2} + a - 3\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 3\) , \( -15 a^{3} + 12 a^{2} + 50 a - 50\) , \( -1451 a^{3} + 1696 a^{2} + 5243 a - 6150\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-15a^{3}+12a^{2}+50a-50\right){x}-1451a^{3}+1696a^{2}+5243a-6150$
64.1-a3 64.1-a \(\Q(\zeta_{20})^+\) \( 2^{6} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $22.99044218$ 1.028163831 \( -2711191688 a^{2} + 9809183676 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{3} + a^{2} - 3 a - 2\) , \( 5 a^{3} - 12 a^{2} - 2 a + 10\) , \( 890 a^{3} - 1696 a^{2} - 1220 a + 2330\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(5a^{3}-12a^{2}-2a+10\right){x}+890a^{3}-1696a^{2}-1220a+2330$
64.1-a4 64.1-a \(\Q(\zeta_{20})^+\) \( 2^{6} \) 0 $\Z/8\Z$ $\mathrm{SU}(2)$ $1$ $1471.388299$ 1.028163831 \( 548896 a^{2} - 757104 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} - a + 4\) , \( a^{2} + a - 3\) , \( 4 a^{3} - 10 a^{2} - 18 a + 31\) , \( 11 a^{3} - 11 a^{2} - 38 a + 43\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(4a^{3}-10a^{2}-18a+31\right){x}+11a^{3}-11a^{2}-38a+43$
64.1-a5 64.1-a \(\Q(\zeta_{20})^+\) \( 2^{6} \) 0 $\Z/2\Z\oplus\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $1471.388299$ 1.028163831 \( 2048 \) \( \bigl[0\) , \( a^{2} - 3\) , \( 0\) , \( -a^{2} + 2\) , \( 0\bigr] \) ${y}^2={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-a^{2}+2\right){x}$
64.1-a6 64.1-a \(\Q(\zeta_{20})^+\) \( 2^{6} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $367.8470749$ 1.028163831 \( 78608 \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( 12 a^{3} + 13 a^{2} - 39 a - 44\) , \( 20 a^{3} + 22 a^{2} - 69 a - 79\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(12a^{3}+13a^{2}-39a-44\right){x}+20a^{3}+22a^{2}-69a-79$
64.1-b1 64.1-b \(\Q(\zeta_{20})^+\) \( 2^{6} \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $0.073302756$ $949.4144128$ 1.556184659 \( 2711191688 a^{2} - 3746774764 \) \( \bigl[a^{2} + a - 3\) , \( -a^{3} + a^{2} + 2 a - 4\) , \( a^{2} + a - 3\) , \( -15 a^{3} + 12 a^{2} + 50 a - 50\) , \( 1450 a^{3} - 1696 a^{2} - 5240 a + 6148\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-4\right){x}^{2}+\left(-15a^{3}+12a^{2}+50a-50\right){x}+1450a^{3}-1696a^{2}-5240a+6148$
64.1-b2 64.1-b \(\Q(\zeta_{20})^+\) \( 2^{6} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.146605513$ $237.3536032$ 1.556184659 \( -548896 a^{2} + 1987376 \) \( \bigl[a^{2} + a - 3\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{3} - 2 a + 1\) , \( 4 a^{3} + 5 a^{2} - 9 a - 9\) , \( -a^{3} - 4 a^{2} - 5 a - 3\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(4a^{3}+5a^{2}-9a-9\right){x}-a^{3}-4a^{2}-5a-3$
64.1-b3 64.1-b \(\Q(\zeta_{20})^+\) \( 2^{6} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.146605513$ $237.3536032$ 1.556184659 \( 548896 a^{2} - 757104 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - 3\) , \( a^{3} - 2 a + 1\) , \( 3 a^{3} - 8 a^{2} - 14 a + 23\) , \( -8 a^{3} + 2 a^{2} + 24 a - 18\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(3a^{3}-8a^{2}-14a+23\right){x}-8a^{3}+2a^{2}+24a-18$
64.1-b4 64.1-b \(\Q(\zeta_{20})^+\) \( 2^{6} \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $0.073302756$ $949.4144128$ 1.556184659 \( -2711191688 a^{2} + 9809183676 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} - a^{2} + 3 a + 4\) , \( 0\) , \( 3 a^{3} - 14 a^{2} + 6 a + 19\) , \( -886 a^{3} + 1683 a^{2} + 1222 a - 2316\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(-a^{3}-a^{2}+3a+4\right){x}^{2}+\left(3a^{3}-14a^{2}+6a+19\right){x}-886a^{3}+1683a^{2}+1222a-2316$
64.1-b5 64.1-b \(\Q(\zeta_{20})^+\) \( 2^{6} \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $0.073302756$ $949.4144128$ 1.556184659 \( 2048 \) \( \bigl[0\) , \( -a^{2} + 3\) , \( 0\) , \( -a^{2} + 2\) , \( 0\bigr] \) ${y}^2={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{2}+2\right){x}$
64.1-b6 64.1-b \(\Q(\zeta_{20})^+\) \( 2^{6} \) $1$ $\Z/2\Z\oplus\Z/4\Z$ $\mathrm{SU}(2)$ $0.146605513$ $3797.657651$ 1.556184659 \( 78608 \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - 4 a - 1\) , \( 0\) , \( 11 a^{3} + 10 a^{2} - 38 a - 38\) , \( -45 a^{3} - 52 a^{2} + 162 a + 190\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(11a^{3}+10a^{2}-38a-38\right){x}-45a^{3}-52a^{2}+162a+190$
  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.