Label
Class
Class size
Class degree
Base field
Field degree
Field signature
Conductor
Conductor norm
Discriminant norm
Root analytic conductor
Bad primes
Rank
Torsion
CM
CM
Sato-Tate
$\Q$-curve
Base change
Semistable
Potentially good
Nonmax $\ell$
mod-$\ell$ images
$Ш_{\textrm{an}}$
Tamagawa
Regulator
Period
Leading coeff
j-invariant
Weierstrass coefficients
Weierstrass equation
32.1-a1
32.1-a
$8$
$16$
\(\Q(\zeta_{16})^+\)
$4$
$[4, 0]$
32.1
\( 2^{5} \)
\( 2^{12} \)
$6.23659$
$(a)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{potential}$
$-16$
$N(\mathrm{U}(1))$
✓
✓
✓
$2$
2Cs
$1$
\( 2^{2} \)
$1$
$189.0727201$
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
$8$
$16$
\(\Q(\zeta_{16})^+\)
$4$
$[4, 0]$
32.1
\( 2^{5} \)
\( 2^{12} \)
$6.23659$
$(a)$
0
$\Z/2\Z\oplus\Z/8\Z$
$\textsf{potential}$
$-16$
$N(\mathrm{U}(1))$
✓
✓
✓
$2$
2Cs
$1$
\( 2^{2} \)
$1$
$3025.163522$
1.044489082
\( 287496 \)
\( \bigl[a^{2} - 2\) , \( 1\) , \( a^{2} - 2\) , \( -3\) , \( 0\bigr] \)
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+{x}^{2}-3{x}$
256.1-e1
256.1-e
$8$
$16$
\(\Q(\zeta_{16})^+\)
$4$
$[4, 0]$
256.1
\( 2^{8} \)
\( 2^{12} \)
$8.08786$
$(a)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{potential}$
$-16$
$N(\mathrm{U}(1))$
✓
✓
✓
$2$
2Cs
$1$
\( 2 \)
$0.304354515$
$756.2908805$
2.543159752
\( 287496 \)
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 6 a^{2} - 2 a - 21\) , \( -10 a^{2} - a + 33\bigr] \)
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(6a^{2}-2a-21\right){x}-10a^{2}-a+33$
256.1-e4
256.1-e
$8$
$16$
\(\Q(\zeta_{16})^+\)
$4$
$[4, 0]$
256.1
\( 2^{8} \)
\( 2^{12} \)
$8.08786$
$(a)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{potential}$
$-16$
$N(\mathrm{U}(1))$
✓
✓
✓
$2$
2Cs
$1$
\( 2 \)
$0.304354515$
$756.2908805$
2.543159752
\( 287496 \)
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{2} - a - 2\) , \( 0\) , \( 7 a^{2} - 20\) , \( 8 a^{2} - 26\bigr] \)
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(7a^{2}-20\right){x}+8a^{2}-26$
Download
displayed columns for
results
to
Text
Pari/GP
SageMath
Magma
Oscar
CSV
*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.