| 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 |
| 11.1-a1 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{8} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$120.9845885$ |
1.147921305 |
\( -\frac{1271382434328458944}{214358881} a^{3} + \frac{3189239753697071944}{214358881} a^{2} + \frac{274621437566108395}{214358881} a - \frac{1685647152341252737}{214358881} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -1\) , \( a^{3} - 4 a\) , \( 5 a^{3} - 14 a^{2} + 3 a + 1\) , \( -16 a^{3} + 40 a^{2} - 7 a - 14\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}-{x}^{2}+\left(5a^{3}-14a^{2}+3a+1\right){x}-16a^{3}+40a^{2}-7a-14$ |
| 11.1-a2 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$241.9691770$ |
1.147921305 |
\( \frac{143460}{11} a^{3} - \frac{313574}{11} a^{2} - \frac{467081}{11} a + \frac{520237}{11} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + 2 a^{2} - a - 1\) , \( a^{3} + 3 a^{2} - 4\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(a^{3}+2a^{2}-a-1\right){x}+a^{3}+3a^{2}-4$ |
| 11.1-a3 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$483.9383541$ |
1.147921305 |
\( \frac{11189839269}{121} a^{3} + \frac{15244624290}{121} a^{2} - \frac{8746346853}{121} a - \frac{9472934785}{121} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{2} - a - 1\) , \( -2 a^{2} - a + 1\) , \( -2 a^{2} + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}-a{x}^{2}+\left(-2a^{2}-a+1\right){x}-2a^{2}+1$ |
| 11.1-a4 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$60.49229427$ |
1.147921305 |
\( \frac{1702129469435805324}{11} a^{3} + \frac{2318878776895144329}{11} a^{2} - \frac{1330538175521684262}{11} a - \frac{1441053861737062367}{11} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{2} - a - 1\) , \( -5 a^{3} - 27 a^{2} + 4 a + 16\) , \( -36 a^{3} - 102 a^{2} + 35 a + 61\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}-a{x}^{2}+\left(-5a^{3}-27a^{2}+4a+16\right){x}-36a^{3}-102a^{2}+35a+61$ |
| 11.1-a5 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{2} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$120.9845885$ |
1.147921305 |
\( -\frac{218889110914391240}{121} a^{3} + \frac{37981189225429456}{121} a^{2} + \frac{907177163042977213}{121} a + \frac{530680189689696129}{121} \) |
\( \bigl[1\) , \( -a^{3} + 3 a\) , \( a^{2} - 1\) , \( -68 a^{3} + 158 a^{2} + 160 a - 355\) , \( 927 a^{3} - 1319 a^{2} - 2859 a + 1883\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-68a^{3}+158a^{2}+160a-355\right){x}+927a^{3}-1319a^{2}-2859a+1883$ |
| 11.1-a6 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( 11^{4} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$483.9383541$ |
1.147921305 |
\( -\frac{261546261312}{14641} a^{3} + \frac{36220688425}{14641} a^{2} + \frac{1091148881914}{14641} a + \frac{668876936913}{14641} \) |
\( \bigl[1\) , \( -a^{3} + 3 a\) , \( a^{2} - 1\) , \( -3 a^{3} + 8 a^{2} + 10 a - 20\) , \( 16 a^{3} - 23 a^{2} - 54 a + 33\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-3a^{3}+8a^{2}+10a-20\right){x}+16a^{3}-23a^{2}-54a+33$ |
*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.
