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-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 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -7\) , \( -5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-7{x}-5$ |
144.1-b5 |
144.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
144.1 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$0.87554$ |
$(a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
1.004711853 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -7\) , \( 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-7{x}+4$ |
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 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -55\) , \( 121\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-55{x}+121$ |
1296.1-b5 |
1296.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{20} \) |
$1.51647$ |
$(a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.158547729$ |
$3.100373158$ |
2.366086572 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -54\) , \( -176\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-54{x}-176$ |
2304.1-c5 |
2304.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{8} \) |
$1.75107$ |
$(a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.070944103$ |
$7.270694035$ |
2.752945917 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -48 a - 72\) , \( 180 a + 252\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-48a-72\right){x}+180a+252$ |
2304.1-s5 |
2304.1-s |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{8} \) |
$1.75107$ |
$(a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.070944103$ |
$7.270694035$ |
2.752945917 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -48 a - 72\) , \( -180 a - 252\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-48a-72\right){x}-180a-252$ |
3528.2-d5 |
3528.2-d |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3528.2 |
\( 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{6} \) |
$1.94790$ |
$(a), (-2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.789108905$ |
$5.496128079$ |
3.066752947 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 24 a - 55\) , \( -99 a + 112\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(24a-55\right){x}-99a+112$ |
3528.3-d5 |
3528.3-d |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3528.3 |
\( 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{6} \) |
$1.94790$ |
$(a), (2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.789108905$ |
$5.496128079$ |
3.066752947 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -24 a - 55\) , \( 99 a + 112\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-24a-55\right){x}+99a+112$ |
*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.