| 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 |
| 99.1-a1 |
99.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{16} \cdot 11^{8} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.291285250$ |
1.324400503 |
\( \frac{98931640625}{96059601} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -98 a + 478\) , \( 745 a - 3625\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-98a+478\right){x}+745a-3625$ |
| 99.1-a2 |
99.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{32} \cdot 11^{4} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.291285250$ |
1.324400503 |
\( \frac{14553591673375}{5208653241} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 507 a - 2547\) , \( 9094 a - 44402\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(507a-2547\right){x}+9094a-44402$ |
| 99.1-b1 |
99.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{24} \cdot 11^{2} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$1.883253066$ |
1.287699630 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -386 a + 1934\) , \( 2836 a - 13797\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-386a+1934\right){x}+2836a-13797$ |
| 99.1-b2 |
99.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{12} \cdot 11^{4} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$7.533012266$ |
1.287699630 |
\( \frac{169112377}{88209} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -99 a - 388\) , \( -804 a - 3130\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-99a-388\right){x}-804a-3130$ |
| 99.1-b3 |
99.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{6} \cdot 11^{2} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$30.13204906$ |
1.287699630 |
\( \frac{30664297}{297} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 64 a - 266\) , \( -484 a + 2427\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(64a-266\right){x}-484a+2427$ |
| 99.1-b4 |
99.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{6} \cdot 11^{8} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$1.883253066$ |
1.287699630 |
\( \frac{347873904937}{395307} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 1324 a - 6426\) , \( 54956 a - 268543\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1324a-6426\right){x}+54956a-268543$ |
| 99.1-c1 |
99.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{24} \cdot 11^{2} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.234063206$ |
1.527570785 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 44\) , \( 55\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+44{x}+55$ |
| 99.1-c2 |
99.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{12} \cdot 11^{4} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$8.936252827$ |
1.527570785 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -11\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-11{x}$ |
| 99.1-c3 |
99.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{6} \cdot 11^{2} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$8.936252827$ |
1.527570785 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -6\) , \( -9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-6{x}-9$ |
| 99.1-c4 |
99.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{6} \cdot 11^{8} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$8.936252827$ |
1.527570785 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -146\) , \( 621\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-146{x}+621$ |
| 99.1-d1 |
99.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{16} \cdot 11^{8} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.291285250$ |
1.324400503 |
\( \frac{98931640625}{96059601} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 96 a + 381\) , \( -746 a - 2880\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(96a+381\right){x}-746a-2880$ |
| 99.1-d2 |
99.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
99.1 |
\( 3^{2} \cdot 11 \) |
\( 3^{32} \cdot 11^{4} \) |
$2.47339$ |
$(a-6), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.291285250$ |
1.324400503 |
\( \frac{14553591673375}{5208653241} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -509 a - 2039\) , \( -9095 a - 35308\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-509a-2039\right){x}-9095a-35308$ |
*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.
