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
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( - 2^{9} \)
$0.60113$
$(a)$
0
$\Z/2\Z$
$\textsf{potential}$
$-64$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 2 \)
$1$
$3.437592909$
0.607686314
\( -29071392966 a + 41113158120 \)
\( \bigl[a\) , \( 1\) , \( 0\) , \( 15 a - 22\) , \( 46 a - 69\bigr] \)
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(15a-22\right){x}+46a-69$
32.1-a2
32.1-a
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( - 2^{9} \)
$0.60113$
$(a)$
0
$\Z/4\Z$
$\textsf{potential}$
$-64$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 1 \)
$1$
$27.50074327$
0.607686314
\( -29071392966 a + 41113158120 \)
\( \bigl[a\) , \( 1\) , \( a\) , \( 15 a - 23\) , \( -31 a + 46\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(15a-23\right){x}-31a+46$
32.1-a3
32.1-a
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( 2^{12} \)
$0.60113$
$(a)$
0
$\Z/4\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$
✓
✓
✓
$1$
\( 2 \)
$1$
$13.75037163$
0.607686314
\( 1728 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \)
${y}^2={x}^{3}+{x}$
32.1-a4
32.1-a
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( 2^{12} \)
$0.60113$
$(a)$
0
$\Z/2\Z\oplus\Z/4\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$
✓
✓
✓
$2$
2Cs
$1$
\( 2^{2} \)
$1$
$27.50074327$
0.607686314
\( 1728 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \)
${y}^2={x}^{3}-{x}$
32.1-a5
32.1-a
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( 2^{6} \)
$0.60113$
$(a)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{potential}$
$-16$
$N(\mathrm{U}(1))$
✓
✓
✓
$2$
2Cs
$1$
\( 2 \)
$1$
$13.75037163$
0.607686314
\( 287496 \)
\( \bigl[a\) , \( 1\) , \( 0\) , \( -2\) , \( -3\bigr] \)
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-2{x}-3$
32.1-a6
32.1-a
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( 2^{6} \)
$0.60113$
$(a)$
0
$\Z/2\Z\oplus\Z/4\Z$
$\textsf{potential}$
$-16$
$N(\mathrm{U}(1))$
✓
✓
✓
$2$
2Cs
$1$
\( 2 \)
$1$
$55.00148654$
0.607686314
\( 287496 \)
\( \bigl[a\) , \( 1\) , \( a\) , \( -3\) , \( 0\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-3{x}$
32.1-a7
32.1-a
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( - 2^{9} \)
$0.60113$
$(a)$
0
$\Z/2\Z$
$\textsf{potential}$
$-64$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 2 \)
$1$
$3.437592909$
0.607686314
\( 29071392966 a + 41113158120 \)
\( \bigl[a\) , \( 1\) , \( 0\) , \( -15 a - 22\) , \( -46 a - 69\bigr] \)
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-15a-22\right){x}-46a-69$
32.1-a8
32.1-a
$8$
$16$
\(\Q(\sqrt{2}) \)
$2$
$[2, 0]$
32.1
\( 2^{5} \)
\( - 2^{9} \)
$0.60113$
$(a)$
0
$\Z/4\Z$
$\textsf{potential}$
$-64$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 1 \)
$1$
$27.50074327$
0.607686314
\( 29071392966 a + 41113158120 \)
\( \bigl[a\) , \( 1\) , \( a\) , \( -15 a - 23\) , \( 31 a + 46\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-15a-23\right){x}+31a+46$
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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.