| 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 |
$2$ |
$2$ |
6.6.966125.1 |
$6$ |
$[6, 0]$ |
11.1 |
\( 11 \) |
\( 11^{4} \) |
$107.26004$ |
$(a^3-3a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$4005.949440$ |
2.03779 |
\( \frac{26409275481}{14641} a^{5} - \frac{37964648544}{14641} a^{4} - \frac{141838839729}{14641} a^{3} + \frac{167713909763}{14641} a^{2} + \frac{137868294632}{14641} a - \frac{60345357700}{14641} \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 5 a^{2} - 2\) , \( -a^{5} + a^{4} + 4 a^{3} - 5 a^{2} - a + 3\) , \( 0\) , \( -a^{2} + a + 2\) , \( 0\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+5a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-5a^{2}-a+3\right){x}^{2}+\left(-a^{2}+a+2\right){x}$ |
| 11.1-a2 |
11.1-a |
$2$ |
$2$ |
6.6.966125.1 |
$6$ |
$[6, 0]$ |
11.1 |
\( 11 \) |
\( 11^{8} \) |
$107.26004$ |
$(a^3-3a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$1001.487360$ |
2.03779 |
\( -\frac{2073398676004049181537}{214358881} a^{5} + \frac{2980851192898443467617}{214358881} a^{4} + \frac{11135780014883593519015}{214358881} a^{3} - \frac{13167327751171658528183}{214358881} a^{2} - \frac{10824320631915953239657}{214358881} a + \frac{4737418469535071469153}{214358881} \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 5 a^{2} - 2\) , \( -a^{5} + a^{4} + 4 a^{3} - 5 a^{2} - a + 3\) , \( 0\) , \( 4 a^{2} - 4 a - 8\) , \( -a^{5} - a^{4} + 6 a^{3} + 14 a^{2} - 16 a - 22\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+5a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-5a^{2}-a+3\right){x}^{2}+\left(4a^{2}-4a-8\right){x}-a^{5}-a^{4}+6a^{3}+14a^{2}-16a-22$ |
| 11.1-b1 |
11.1-b |
$1$ |
$1$ |
6.6.966125.1 |
$6$ |
$[6, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$107.26004$ |
$(a^3-3a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$1501.744202$ |
1.52785 |
\( \frac{104855673}{11} a^{5} + \frac{213667626}{11} a^{4} - \frac{470839075}{11} a^{3} - \frac{1001999556}{11} a^{2} - \frac{19437862}{11} a + \frac{137797178}{11} \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + a\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 9 a\) , \( a^{3} - 4 a\) , \( 2 a^{5} + a^{4} - 9 a^{3} - 4 a^{2} + 2 a + 3\) , \( 2 a^{5} + a^{4} - 8 a^{3} - 3 a^{2}\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+a\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-9a\right){x}^{2}+\left(2a^{5}+a^{4}-9a^{3}-4a^{2}+2a+3\right){x}+2a^{5}+a^{4}-8a^{3}-3a^{2}$ |
| 11.1-c1 |
11.1-c |
$1$ |
$1$ |
6.6.966125.1 |
$6$ |
$[6, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$107.26004$ |
$(a^3-3a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.004399879$ |
$85413.69335$ |
2.29405 |
\( \frac{104855673}{11} a^{5} + \frac{213667626}{11} a^{4} - \frac{470839075}{11} a^{3} - \frac{1001999556}{11} a^{2} - \frac{19437862}{11} a + \frac{137797178}{11} \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 2\) , \( -a^{5} + a^{4} + 4 a^{3} - 5 a^{2} + a + 3\) , \( 2 a^{5} - 3 a^{4} - 9 a^{3} + 13 a^{2} + 4 a - 2\) , \( -a^{5} + 5 a^{3} - 5 a\) , \( -4 a^{5} + 6 a^{4} + 19 a^{3} - 25 a^{2} - 11 a + 2\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-2\right){x}{y}+\left(2a^{5}-3a^{4}-9a^{3}+13a^{2}+4a-2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-5a^{2}+a+3\right){x}^{2}+\left(-a^{5}+5a^{3}-5a\right){x}-4a^{5}+6a^{4}+19a^{3}-25a^{2}-11a+2$ |
| 11.1-d1 |
11.1-d |
$2$ |
$2$ |
6.6.966125.1 |
$6$ |
$[6, 0]$ |
11.1 |
\( 11 \) |
\( 11^{4} \) |
$107.26004$ |
$(a^3-3a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.004439684$ |
$69591.14624$ |
1.88600 |
\( \frac{26409275481}{14641} a^{5} - \frac{37964648544}{14641} a^{4} - \frac{141838839729}{14641} a^{3} + \frac{167713909763}{14641} a^{2} + \frac{137868294632}{14641} a - \frac{60345357700}{14641} \) |
\( \bigl[a^{4} - 4 a^{2} + a\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - 1\) , \( 0\) , \( -a^{4} + 4 a^{2}\) , \( 0\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+a\right){x}{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-1\right){x}^{2}+\left(-a^{4}+4a^{2}\right){x}$ |
| 11.1-d2 |
11.1-d |
$2$ |
$2$ |
6.6.966125.1 |
$6$ |
$[6, 0]$ |
11.1 |
\( 11 \) |
\( 11^{8} \) |
$107.26004$ |
$(a^3-3a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.008879369$ |
$17397.78656$ |
1.88600 |
\( -\frac{2073398676004049181537}{214358881} a^{5} + \frac{2980851192898443467617}{214358881} a^{4} + \frac{11135780014883593519015}{214358881} a^{3} - \frac{13167327751171658528183}{214358881} a^{2} - \frac{10824320631915953239657}{214358881} a + \frac{4737418469535071469153}{214358881} \) |
\( \bigl[a^{4} - 4 a^{2} + a\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - 1\) , \( 0\) , \( 4 a^{4} - 16 a^{2}\) , \( 6 a^{5} - 7 a^{4} - 25 a^{3} + 29 a^{2} + 3 a - 1\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+a\right){x}{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-1\right){x}^{2}+\left(4a^{4}-16a^{2}\right){x}+6a^{5}-7a^{4}-25a^{3}+29a^{2}+3a-1$ |
*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.
