| 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 |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 7 \) |
$0.134327059$ |
$6.120386052$ |
4.907824392 |
\( -\frac{1383171944}{4782969} a + \frac{1773386828}{4782969} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -1211 a - 5681\) , \( 80213 a + 376233\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-1211a-5681\right){x}+80213a+376233$ |
| 72.1-a2 |
72.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{8} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$0.268654118$ |
$24.48154420$ |
4.907824392 |
\( \frac{608455232}{2187} a + \frac{2857688944}{2187} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -1421 a - 6666\) , \( 60522 a + 283874\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-1421a-6666\right){x}+60522a+283874$ |
| 72.1-b1 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$5.683508517$ |
1.211728087 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 7\) , \( 35\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+7{x}+35$ |
| 72.1-b2 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.36701703$ |
1.211728087 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 11032 a + 51745\) , \( -1694014 a - 7945630\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(11032a+51745\right){x}-1694014a-7945630$ |
| 72.1-b3 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
1.211728087 |
\( \frac{35152}{9} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 2\) , \( 2\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+2{x}+2$ |
| 72.1-b4 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
1.211728087 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -3\) , \( -3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-3{x}-3$ |
| 72.1-b5 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$5.683508517$ |
1.211728087 |
\( \frac{28756228}{3} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -13\) , \( -55\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-13{x}-55$ |
| 72.1-b6 |
72.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
1.211728087 |
\( \frac{3065617154}{9} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -93\) , \( 159\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-93{x}+159$ |
| 72.1-c1 |
72.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 7 \) |
$1$ |
$2.587772881$ |
3.862005225 |
\( \frac{1383171944}{4782969} a + \frac{1773386828}{4782969} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 168 a + 798\) , \( 252 a + 1188\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(168a+798\right){x}+252a+1188$ |
| 72.1-c2 |
72.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{8} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$10.35109152$ |
3.862005225 |
\( -\frac{608455232}{2187} a + \frac{2857688944}{2187} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -42 a - 187\) , \( -84 a - 388\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-42a-187\right){x}-84a-388$ |
| 72.1-d1 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.832319552$ |
$2.325279868$ |
2.808252391 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -64813 a + 304014\) , \( 142270436 a - 667307485\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-64813a+304014\right){x}+142270436a-667307485$ |
| 72.1-d2 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$5.664639104$ |
$18.60223895$ |
2.808252391 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+{x}$ |
| 72.1-d3 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.832319552$ |
$37.20447790$ |
2.808252391 |
\( \frac{35152}{9} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 17927 a - 84071\) , \( -2074079 a + 9728303\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(17927a-84071\right){x}-2074079a+9728303$ |
| 72.1-d4 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.416159776$ |
$9.301119475$ |
2.808252391 |
\( \frac{1556068}{81} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 100667 a - 472156\) , \( 36137766 a - 169501137\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(100667a-472156\right){x}+36137766a-169501137$ |
| 72.1-d5 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.416159776$ |
$37.20447790$ |
2.808252391 |
\( \frac{28756228}{3} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -266147 a - 1248326\) , \( 161273084 a + 756437825\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-266147a-1248326\right){x}+161273084a+756437825$ |
| 72.1-d6 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.832319552$ |
$2.325279868$ |
2.808252391 |
\( \frac{3065617154}{9} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 1589987 a - 7457686\) , \( 2366958216 a - 11102018109\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(1589987a-7457686\right){x}+2366958216a-11102018109$ |
| 72.1-e1 |
72.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 7 \) |
$0.134327059$ |
$6.120386052$ |
4.907824392 |
\( \frac{1383171944}{4782969} a + \frac{1773386828}{4782969} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 1211 a - 5681\) , \( -80213 a + 376233\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(1211a-5681\right){x}-80213a+376233$ |
| 72.1-e2 |
72.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{8} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$0.268654118$ |
$24.48154420$ |
4.907824392 |
\( -\frac{608455232}{2187} a + \frac{2857688944}{2187} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 1421 a - 6666\) , \( -60522 a + 283874\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(1421a-6666\right){x}-60522a+283874$ |
| 72.1-f1 |
72.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 7 \) |
$1$ |
$2.587772881$ |
3.862005225 |
\( -\frac{1383171944}{4782969} a + \frac{1773386828}{4782969} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -168 a + 798\) , \( -252 a + 1188\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-168a+798\right){x}-252a+1188$ |
| 72.1-f2 |
72.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{22}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{8} \) |
$2.44182$ |
$(-3a-14), (-a+5), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$10.35109152$ |
3.862005225 |
\( \frac{608455232}{2187} a + \frac{2857688944}{2187} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 42 a - 187\) , \( 84 a - 388\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(42a-187\right){x}+84a-388$ |
*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.
