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
31.1-a1
31.1-a
$4$
$4$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( - 31^{4} \)
$0.59638$
$(-4a-1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \)
$1$
$2.544454463$
0.449800251
\( -\frac{78588372777605}{923521} a + \frac{111138046783591}{923521} \)
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -7 a - 21\) , \( -31 a - 52\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7a-21\right){x}-31a-52$
31.1-a2
31.1-a
$4$
$4$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( -31 \)
$0.59638$
$(-4a-1)$
0
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 1 \)
$1$
$20.35563570$
0.449800251
\( \frac{47780}{31} a + \frac{69151}{31} \)
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -2 a - 1\) , \( -a - 2\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-1\right){x}-a-2$
31.1-a3
31.1-a
$4$
$4$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( 31^{2} \)
$0.59638$
$(-4a-1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2 \)
$1$
$10.17781785$
0.449800251
\( \frac{4423034250}{961} a + \frac{6270751283}{961} \)
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 10 a - 14\) , \( 21 a - 30\bigr] \)
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a-14\right){x}+21a-30$
31.1-a4
31.1-a
$4$
$4$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
31.1
\( 31 \)
\( 31 \)
$0.59638$
$(-4a-1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 1 \)
$1$
$5.088908926$
0.449800251
\( \frac{1861812264958875}{31} a + \frac{2633000155833143}{31} \)
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 25 a - 49\) , \( -79 a + 100\bigr] \)
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(25a-49\right){x}-79a+100$
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Pari/GP
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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.