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 |
72.2-a1 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$1.817673508$ |
0.642644632 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 5\) , \( 23\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+5{x}+23$ |
72.2-a2 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.270694035$ |
0.642644632 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+{x}$ |
72.2-a3 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.270694035$ |
0.642644632 |
\( \frac{35152}{9} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}$ |
72.2-a4 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$3.635347017$ |
0.642644632 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-5{x}+5$ |
72.2-a5 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{11} \cdot 3^{20} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.908836754$ |
0.642644632 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 70 a + 85\) , \( -98 a + 559\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(70a+85\right){x}-98a+559$ |
72.2-a6 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{11} \cdot 3^{20} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.908836754$ |
0.642644632 |
\( \frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -70 a + 85\) , \( 98 a + 559\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-70a+85\right){x}+98a+559$ |
72.2-a7 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.635347017$ |
0.642644632 |
\( \frac{28756228}{3} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -15\) , \( -27\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-15{x}-27$ |
72.2-a8 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.817673508$ |
0.642644632 |
\( \frac{3065617154}{9} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -95\) , \( 347\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-95{x}+347$ |
*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.