| 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 |
| 648.1-a1 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{11} \cdot 3^{14} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.789005678$ |
1.339615804 |
\( -\frac{123062343233293457}{3} a + 58012144939294440 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 4590 a - 7345\) , \( -215622 a + 314563\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(4590a-7345\right){x}-215622a+314563$ |
| 648.1-a2 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{28} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.894502839$ |
1.339615804 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 35\) , \( 607\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+35{x}+607$ |
| 648.1-a3 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{14} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.789005678$ |
1.339615804 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -7\bigr] \) |
${y}^2={x}^{3}+6{x}-7$ |
| 648.1-a4 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{16} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.578011356$ |
1.339615804 |
\( \frac{35152}{9} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -10\) , \( -14\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-10{x}-14$ |
| 648.1-a5 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{20} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.578011356$ |
1.339615804 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -55\) , \( 121\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-55{x}+121$ |
| 648.1-a6 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{14} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.894502839$ |
1.339615804 |
\( \frac{28756228}{3} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -145\) , \( -743\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-145{x}-743$ |
| 648.1-a7 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{16} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.578011356$ |
1.339615804 |
\( \frac{3065617154}{9} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -865\) , \( 9355\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-865{x}+9355$ |
| 648.1-a8 |
648.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{11} \cdot 3^{14} \) |
$1.27520$ |
$(a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.789005678$ |
1.339615804 |
\( \frac{123062343233293457}{3} a + 58012144939294440 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -4590 a - 7345\) , \( 215622 a + 314563\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-4590a-7345\right){x}+215622a+314563$ |
*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.
