| 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 |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{4} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.209864983$ |
$1457.337013$ |
2.55699535 |
\( -\frac{64801139993982903601201}{14641} a^{4} + \frac{79742135469152888974129}{14641} a^{3} + \frac{290678984684285122184960}{14641} a^{2} - \frac{292899070194533892815349}{14641} a - \frac{157977896987517584455026}{14641} \) |
\( \bigl[a^{4} - 4 a^{2} + 2\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 7 a - 4\) , \( a\) , \( -291 a^{4} + 574 a^{3} + 610 a^{2} - 923 a - 458\) , \( 7024 a^{4} - 13684 a^{3} - 15498 a^{2} + 23224 a + 10852\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+2\right){x}{y}+a{y}={x}^{3}+\left(-a^{4}+2a^{3}+4a^{2}-7a-4\right){x}^{2}+\left(-291a^{4}+574a^{3}+610a^{2}-923a-458\right){x}+7024a^{4}-13684a^{3}-15498a^{2}+23224a+10852$ |
| 11.1-a2 |
11.1-a |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{8} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.419729966$ |
$2914.674027$ |
2.55699535 |
\( -\frac{194960202826935862}{214358881} a^{4} + \frac{240211991646390234}{214358881} a^{3} + \frac{874044186101463078}{214358881} a^{2} - \frac{882364691646287529}{214358881} a - \frac{473469439417993593}{214358881} \) |
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 3 a + 6\) , \( a^{4} - 4 a^{2} + a + 3\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 5\) , \( 4 a^{4} + 23 a^{3} + 4 a^{2} - 93 a - 88\) , \( 45 a^{4} + 51 a^{3} - 187 a^{2} - 224 a - 1\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-5a^{2}+3a+6\right){x}{y}+\left(a^{4}-a^{3}-5a^{2}+3a+5\right){y}={x}^{3}+\left(a^{4}-4a^{2}+a+3\right){x}^{2}+\left(4a^{4}+23a^{3}+4a^{2}-93a-88\right){x}+45a^{4}+51a^{3}-187a^{2}-224a-1$ |
| 11.1-a3 |
11.1-a |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{4} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.209864983$ |
$5829.348054$ |
2.55699535 |
\( \frac{199999900}{14641} a^{4} - \frac{195574728}{14641} a^{3} - \frac{840036139}{14641} a^{2} + \frac{774782742}{14641} a + \frac{392874921}{14641} \) |
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 3 a + 6\) , \( a^{4} - 4 a^{2} + a + 3\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 5\) , \( -a^{4} - 2 a^{3} + 4 a^{2} + 12 a + 7\) , \( 4 a^{4} + 5 a^{3} - 15 a^{2} - 20 a - 3\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-5a^{2}+3a+6\right){x}{y}+\left(a^{4}-a^{3}-5a^{2}+3a+5\right){y}={x}^{3}+\left(a^{4}-4a^{2}+a+3\right){x}^{2}+\left(-a^{4}-2a^{3}+4a^{2}+12a+7\right){x}+4a^{4}+5a^{3}-15a^{2}-20a-3$ |
| 11.1-a4 |
11.1-a |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{16} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$0.839459932$ |
$91.08356335$ |
2.55699535 |
\( \frac{476417640375476377168321}{45949729863572161} a^{4} - \frac{921552595192807794564289}{45949729863572161} a^{3} - \frac{1018805365320562990606080}{45949729863572161} a^{2} + \frac{1637467232063013399273925}{45949729863572161} a + \frac{751306184458928265884514}{45949729863572161} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 6\) , \( -34 a^{4} - 106 a^{3} - 82 a^{2} - 9 a - 5\) , \( -1108 a^{4} - 2689 a^{3} + 484 a^{2} + 3592 a + 1217\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{4}-a^{3}-5a^{2}+3a+6\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-34a^{4}-106a^{3}-82a^{2}-9a-5\right){x}-1108a^{4}-2689a^{3}+484a^{2}+3592a+1217$ |
| 11.1-b1 |
11.1-b |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{4} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$18.28997238$ |
1.87789849 |
\( -\frac{64801139993982903601201}{14641} a^{4} + \frac{79742135469152888974129}{14641} a^{3} + \frac{290678984684285122184960}{14641} a^{2} - \frac{292899070194533892815349}{14641} a - \frac{157977896987517584455026}{14641} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( 0\) , \( 5 a^{4} + 67 a^{3} - 14 a^{2} - 226 a - 146\) , \( 68 a^{4} + 285 a^{3} - 223 a^{2} - 1067 a - 602\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-3\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(5a^{4}+67a^{3}-14a^{2}-226a-146\right){x}+68a^{4}+285a^{3}-223a^{2}-1067a-602$ |
| 11.1-b2 |
11.1-b |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{8} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
|
\( 2 \) |
$1$ |
$585.2791162$ |
1.87789849 |
\( -\frac{194960202826935862}{214358881} a^{4} + \frac{240211991646390234}{214358881} a^{3} + \frac{874044186101463078}{214358881} a^{2} - \frac{882364691646287529}{214358881} a - \frac{473469439417993593}{214358881} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 3\) , \( a^{3} - a^{2} - 3 a\) , \( 0\) , \( 12 a^{3} - 4 a^{2} - 36 a - 16\) , \( 6 a^{4} - 47 a^{3} + 4 a^{2} + 119 a + 39\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-3\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(12a^{3}-4a^{2}-36a-16\right){x}+6a^{4}-47a^{3}+4a^{2}+119a+39$ |
| 11.1-b3 |
11.1-b |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{4} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$0 \le r \le 1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$1170.558232$ |
1.87789849 |
\( \frac{199999900}{14641} a^{4} - \frac{195574728}{14641} a^{3} - \frac{840036139}{14641} a^{2} + \frac{774782742}{14641} a + \frac{392874921}{14641} \) |
\( \bigl[a^{2} - 1\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 7\) , \( a^{4} - 4 a^{2} + 3\) , \( a^{4} - 2 a^{3} - 5 a^{2} + 6 a + 7\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 5 a\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{4}-4a^{2}+3\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+5a^{2}-6a-7\right){x}^{2}+\left(a^{4}-2a^{3}-5a^{2}+6a+7\right){x}+a^{4}-2a^{3}-3a^{2}+5a$ |
| 11.1-b4 |
11.1-b |
$4$ |
$4$ |
5.5.89417.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{16} \) |
$33.96158$ |
$(a^3-a^2-3a+2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$292.6395581$ |
1.87789849 |
\( \frac{476417640375476377168321}{45949729863572161} a^{4} - \frac{921552595192807794564289}{45949729863572161} a^{3} - \frac{1018805365320562990606080}{45949729863572161} a^{2} + \frac{1637467232063013399273925}{45949729863572161} a + \frac{751306184458928265884514}{45949729863572161} \) |
\( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{4} - a^{3} - 4 a^{2} + 4 a + 3\) , \( a^{4} - 5 a^{2} + 4\) , \( 19 a^{4} - 67 a^{3} - 120 a^{2} + 207 a + 88\) , \( -36 a^{4} - 539 a^{3} - 321 a^{2} + 1448 a + 587\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+a+3\right){x}{y}+\left(a^{4}-5a^{2}+4\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+4a+3\right){x}^{2}+\left(19a^{4}-67a^{3}-120a^{2}+207a+88\right){x}-36a^{4}-539a^{3}-321a^{2}+1448a+587$ |
*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.
