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 |
72.1-a1 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( - 2^{11} \cdot 3^{2} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$0.581319967$ |
0.822110581 |
\( -\frac{123062343233293457}{3} a + 58012144939294440 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 510 a - 817\) , \( 7986 a - 11651\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(510a-817\right){x}+7986a-11651$ |
72.1-a2 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.325279868$ |
0.822110581 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 3\) , \( -23\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+3{x}-23$ |
72.1-a3 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.60223895$ |
0.822110581 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+{x}$ |
72.1-a4 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$37.20447790$ |
0.822110581 |
\( \frac{35152}{9} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -2\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-2{x}$ |
72.1-a5 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$9.301119475$ |
0.822110581 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -7\) , \( -5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-7{x}-5$ |
72.1-a6 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$37.20447790$ |
0.822110581 |
\( \frac{28756228}{3} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -17\) , \( 27\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-17{x}+27$ |
72.1-a7 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$2.325279868$ |
0.822110581 |
\( \frac{3065617154}{9} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -97\) , \( -347\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-97{x}-347$ |
72.1-a8 |
72.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( - 2^{11} \cdot 3^{2} \) |
$0.73624$ |
$(a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$0.581319967$ |
0.822110581 |
\( \frac{123062343233293457}{3} a + 58012144939294440 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -510 a - 817\) , \( -7986 a - 11651\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-510a-817\right){x}-7986a-11651$ |
*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.