| 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 |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$1.72662$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.808637213$ |
$2.325279868$ |
2.267736573 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -231 a + 793\) , \( 26226 a - 86955\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-231a+793\right){x}+26226a-86955$ |
| 72.1-a2 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$1.72662$ |
$(a+3), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.617274427$ |
$18.60223895$ |
2.267736573 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+{x}$ |
| 72.1-a3 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$1.72662$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.234548855$ |
$37.20447790$ |
2.267736573 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -62 a - 208\) , \( 197 a + 652\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-62a-208\right){x}+197a+652$ |
| 72.1-a4 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$1.72662$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.617274427$ |
$9.301119475$ |
2.267736573 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 369 a - 1197\) , \( 6480 a - 21465\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(369a-1197\right){x}+6480a-21465$ |
| 72.1-a5 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$1.72662$ |
$(a+3), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.617274427$ |
$37.20447790$ |
2.267736573 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 969 a - 3187\) , \( -30024 a + 99605\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(969a-3187\right){x}-30024a+99605$ |
| 72.1-a6 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$1.72662$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.234548855$ |
$2.325279868$ |
2.267736573 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 5769 a - 19107\) , \( 432054 a - 1432935\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5769a-19107\right){x}+432054a-1432935$ |
| 72.1-b1 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$1.72662$ |
$(a+3), (3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$5.683508517$ |
0.856821147 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -234 a + 783\) , \( -25911 a + 85945\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-234a+783\right){x}-25911a+85945$ |
| 72.1-b2 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$1.72662$ |
$(a+3), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$11.36701703$ |
0.856821147 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+{x}$ |
| 72.1-b3 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$1.72662$ |
$(a+3), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$22.73403407$ |
0.856821147 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -63 a - 206\) , \( -537 a - 1782\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-63a-206\right){x}-537a-1782$ |
| 72.1-b4 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$1.72662$ |
$(a+3), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$22.73403407$ |
0.856821147 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 366 a - 1207\) , \( -6955 a + 23075\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(366a-1207\right){x}-6955a+23075$ |
| 72.1-b5 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$1.72662$ |
$(a+3), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.683508517$ |
0.856821147 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 966 a - 3197\) , \( 28759 a - 95375\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(966a-3197\right){x}+28759a-95375$ |
| 72.1-b6 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$1.72662$ |
$(a+3), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$22.73403407$ |
0.856821147 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 5766 a - 19117\) , \( -439639 a + 1458125\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5766a-19117\right){x}-439639a+1458125$ |
*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.
