| 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 |
| 1089.1-a1 |
1089.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{4} \cdot 11^{2} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.122653276$ |
$26.15066734$ |
1.434421970 |
\( -\frac{102485}{33} a - \frac{187018}{99} \) |
\( \bigl[\phi\) , \( \phi\) , \( 1\) , \( -\phi\) , \( 0\bigr] \) |
${y}^2+\phi{x}{y}+{y}={x}^{3}+\phi{x}^{2}-\phi{x}$ |
| 1089.1-a2 |
1089.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{2} \cdot 11^{4} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.061326638$ |
$26.15066734$ |
1.434421970 |
\( \frac{3151430963}{121} a + \frac{5850146561}{363} \) |
\( \bigl[1\) , \( \phi - 1\) , \( 1\) , \( 14 \phi - 29\) , \( -44 \phi + 75\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(14\phi-29\right){x}-44\phi+75$ |
| 1089.1-b1 |
1089.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{4} \cdot 11^{2} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.122653276$ |
$26.15066734$ |
1.434421970 |
\( \frac{102485}{33} a - \frac{494473}{99} \) |
\( \bigl[\phi + 1\) , \( \phi + 1\) , \( 0\) , \( 4 \phi\) , \( \phi + 2\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(\phi+1\right){x}^{2}+4\phi{x}+\phi+2$ |
| 1089.1-b2 |
1089.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{2} \cdot 11^{4} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.061326638$ |
$26.15066734$ |
1.434421970 |
\( -\frac{3151430963}{121} a + \frac{15304439450}{363} \) |
\( \bigl[1\) , \( -\phi\) , \( 1\) , \( -14 \phi - 15\) , \( 44 \phi + 31\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-\phi{x}^{2}+\left(-14\phi-15\right){x}+44\phi+31$ |
| 1089.1-c1 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{24} \cdot 11^{2} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.213608727$ |
$2.234063206$ |
1.280503287 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 44\) , \( 55\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+44{x}+55$ |
| 1089.1-c2 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{12} \cdot 11^{4} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.106804363$ |
$8.936252827$ |
1.280503287 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -11\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-11{x}$ |
| 1089.1-c3 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{2} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.213608727$ |
$8.936252827$ |
1.280503287 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -6\) , \( -9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-6{x}-9$ |
| 1089.1-c4 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{8} \) |
$1.14784$ |
$(-3a+2), (-3a+1), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.053402181$ |
$8.936252827$ |
1.280503287 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -146\) , \( 621\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-146{x}+621$ |
*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.
