| 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 |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.197607477$ |
$16.56670177$ |
3.631848661 |
\( \frac{13072969}{891} a - \frac{29004700}{891} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -2\) , \( -a - 1\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}-2{x}-a-1$ |
| 1089.1-a2 |
1089.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 11^{4} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.395214954$ |
$16.56670177$ |
3.631848661 |
\( -\frac{829562110871}{1089} a + \frac{1910298012317}{1089} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( 15 a - 47\) , \( -46 a + 98\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15a-47\right){x}-46a+98$ |
| 1089.1-b1 |
1089.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.197607477$ |
$16.56670177$ |
3.631848661 |
\( -\frac{13072969}{891} a - \frac{5310577}{297} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -2 a - 1\) , \( -1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-2a-1\right){x}-1$ |
| 1089.1-b2 |
1089.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 11^{4} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.395214954$ |
$16.56670177$ |
3.631848661 |
\( \frac{829562110871}{1089} a + \frac{360245300482}{363} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -17 a - 31\) , \( 45 a + 53\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-17a-31\right){x}+45a+53$ |
| 1089.1-c1 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.37399948$ |
1.854640035 |
\( -\frac{180971}{891} a - \frac{75100}{891} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -2 a + 5\) , \( 6 a - 16\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-2a+5\right){x}+6a-16$ |
| 1089.1-c2 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 11^{8} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.343499872$ |
1.854640035 |
\( -\frac{2608976708689}{131769} a + \frac{2002720393882}{43923} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -97 a - 137\) , \( 141 a + 174\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-97a-137\right){x}+141a+174$ |
| 1089.1-c3 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{4} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$13.37399948$ |
1.854640035 |
\( \frac{3504490381}{9801} a + \frac{1551966373}{3267} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 23 a - 55\) , \( 67 a - 157\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(23a-55\right){x}+67a-157$ |
| 1089.1-c4 |
1089.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.37399948$ |
1.854640035 |
\( \frac{33708003796936883}{72171} a + \frac{14637988882251266}{24057} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 78 a - 220\) , \( -549 a + 1328\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(78a-220\right){x}-549a+1328$ |
| 1089.1-d1 |
1089.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{24} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.184501454$ |
$2.234063206$ |
1.467876014 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 44\) , \( 55\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+44{x}+55$ |
| 1089.1-d2 |
1089.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{12} \cdot 11^{4} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.592250727$ |
$8.936252827$ |
1.467876014 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -11\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-11{x}$ |
| 1089.1-d3 |
1089.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.184501454$ |
$8.936252827$ |
1.467876014 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -6\) , \( -9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-6{x}-9$ |
| 1089.1-d4 |
1089.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{8} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.184501454$ |
$8.936252827$ |
1.467876014 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -146\) , \( 621\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-146{x}+621$ |
| 1089.1-e1 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.37399948$ |
1.854640035 |
\( \frac{180971}{891} a - \frac{85357}{297} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( a + 4\) , \( -7 a - 9\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a+4\right){x}-7a-9$ |
| 1089.1-e2 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 11^{2} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.37399948$ |
1.854640035 |
\( -\frac{33708003796936883}{72171} a + \frac{77621970443690681}{72171} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -79 a - 141\) , \( 548 a + 780\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-79a-141\right){x}+548a+780$ |
| 1089.1-e3 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{4} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$13.37399948$ |
1.854640035 |
\( -\frac{3504490381}{9801} a + \frac{8160389500}{9801} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -24 a - 31\) , \( -68 a - 89\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-24a-31\right){x}-68a-89$ |
| 1089.1-e4 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 11^{8} \) |
$1.85083$ |
$(-a), (-a+1), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.343499872$ |
1.854640035 |
\( \frac{2608976708689}{131769} a + \frac{3399184472957}{131769} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 99 a - 238\) , \( -379 a + 849\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(99a-238\right){x}-379a+849$ |
*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.
