| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{2} \cdot 11^{9} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.400415428$ |
$7.134994802$ |
6.278942589 |
\( \frac{8124934469}{643076643} a - \frac{11191296679}{214358881} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( a + 5\) , \( 32 a + 102\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(a+5\right){x}+32a+102$ |
| 1089.1-b1 |
1089.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{2} \cdot 11^{9} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.400415428$ |
$7.134994802$ |
6.278942589 |
\( -\frac{8124934469}{643076643} a - \frac{25448955568}{643076643} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -a + 6\) , \( -32 a + 134\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+6\right){x}-32a+134$ |
| 1089.1-c1 |
1089.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{3} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.877531895$ |
$2.556432244$ |
4.930366969 |
\( -\frac{78570203}{9801} a - \frac{82303487}{3267} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -4 a - 7\) , \( -4 a - 11\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-7\right){x}-4a-11$ |
| 1089.1-d1 |
1089.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{3} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.877531895$ |
$2.556432244$ |
4.930366969 |
\( \frac{78570203}{9801} a - \frac{325480664}{9801} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 4 a - 11\) , \( 4 a - 15\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(4a-11\right){x}+4a-15$ |
| 1089.1-e1 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{24} \cdot 11^{2} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.893418538$ |
$2.234063206$ |
5.327457962 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 44\) , \( 55\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+44{x}+55$ |
| 1089.1-e2 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{12} \cdot 11^{4} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.446709269$ |
$8.936252827$ |
5.327457962 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -11\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-11{x}$ |
| 1089.1-e3 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{2} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$2.893418538$ |
$8.936252827$ |
5.327457962 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -6\) , \( -9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-6{x}-9$ |
| 1089.1-e4 |
1089.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{8} \) |
$3.73709$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.723354634$ |
$8.936252827$ |
5.327457962 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -146\) , \( 621\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-146{x}+621$ |
*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.
