| 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-c3 |
72.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$1.37737$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$37.20447790$ |
0.878873180 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -50 a - 131\) , \( 146 a + 386\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-50a-131\right){x}+146a+386$ |
| 72.1-d3 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$1.37737$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
2.148164301 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -51 a - 132\) , \( -333 a - 882\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-51a-132\right){x}-333a-882$ |
| 432.1-c3 |
432.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{10} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$16.79094863$ |
1.586595513 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 3 a - 15\) , \( 9 a - 27\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3a-15\right){x}+9a-27$ |
| 432.1-g3 |
432.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{10} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.286163271$ |
$16.79094863$ |
4.081241752 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 5 a - 10\) , \( -5 a + 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(5a-10\right){x}-5a+14$ |
| 432.2-c3 |
432.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{10} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$16.79094863$ |
1.586595513 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -5 a - 15\) , \( -10 a - 27\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5a-15\right){x}-10a-27$ |
| 432.2-g3 |
432.2-g |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{10} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.286163271$ |
$16.79094863$ |
4.081241752 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -5 a - 10\) , \( 5 a + 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-5a-10\right){x}+5a+14$ |
| 648.1-f3 |
648.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{16} \) |
$2.38568$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.409438577$ |
$12.40149263$ |
3.838342240 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -465 a - 1230\) , \( 5609 a + 14840\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-465a-1230\right){x}+5609a+14840$ |
| 648.1-m3 |
648.1-m |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{16} \) |
$2.38568$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.645289777$ |
$7.578011356$ |
3.696502570 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -466 a - 1234\) , \( -7779 a - 20582\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-466a-1234\right){x}-7779a-20582$ |
*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.
