| 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 |
| 27225.7-a1 |
27225.7-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{6} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.697491699$ |
$0.572804391$ |
3.854774863 |
\( \frac{393194}{11} a - \frac{16965365}{11} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -114 a - 416\) , \( -1269 a - 3143\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-114a-416\right){x}-1269a-3143$ |
| 27225.7-a2 |
27225.7-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{6} \cdot 11^{7} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.174372924$ |
$1.145608783$ |
3.854774863 |
\( \frac{7136}{11} a + \frac{4759}{11} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -4 a - 31\) , \( -26 a - 30\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-4a-31\right){x}-26a-30$ |
| 27225.7-b1 |
27225.7-b |
$2$ |
$11$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{10} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$11$ |
11B.10.4[2] |
|
\( 2 \) |
$1$ |
$0.219357806$ |
5.612608756 |
\( -24729001 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2313 a + 4294\) , \( 19720 a + 119193\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2313a+4294\right){x}+19720a+119193$ |
| 27225.7-b2 |
27225.7-b |
$2$ |
$11$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{2} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$11$ |
11B.10.2[2] |
|
\( 2 \) |
$1$ |
$2.412935876$ |
5.612608756 |
\( -121 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3 a + 4\) , \( -2 a - 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-3a+4\right){x}-2a-9$ |
| 27225.7-c1 |
27225.7-c |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 5^{2} \cdot 11^{10} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.692633907$ |
$0.748828780$ |
2.502130266 |
\( -\frac{94706}{363} a + \frac{18753}{11} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -29 a - 54\) , \( -64 a - 104\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-29a-54\right){x}-64a-104$ |
| 27225.7-d1 |
27225.7-d |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 5^{8} \cdot 11^{10} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$0.334886411$ |
1.615552834 |
\( -\frac{94706}{363} a + \frac{18753}{11} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -184 a + 365\) , \( -523 a + 2070\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-184a+365\right){x}-523a+2070$ |
| 27225.7-e1 |
27225.7-e |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$47.86298431$ |
$0.028828495$ |
6.656491569 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -602166 a + 1118307\) , \( 81612670 a + 506323495\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-602166a+1118307\right){x}+81612670a+506323495$ |
| 27225.7-e2 |
27225.7-e |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{16} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5Cs.4.1 |
$1$ |
\( 2^{2} \) |
$9.572596863$ |
$0.144142475$ |
6.656491569 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -796 a + 1477\) , \( 6640 a + 43765\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-796a+1477\right){x}+6640a+43765$ |
| 27225.7-e3 |
27225.7-e |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$1.914519372$ |
$0.720712377$ |
6.656491569 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -26 a + 47\) , \( -70 a - 345\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-26a+47\right){x}-70a-345$ |
| 27225.7-f1 |
27225.7-f |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{13} \cdot 5^{8} \cdot 11^{7} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$4.972725717$ |
$0.327529135$ |
7.857205002 |
\( \frac{2252642504}{601425} a - \frac{714226781}{200475} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -293 a - 185\) , \( -3339 a + 1912\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-293a-185\right){x}-3339a+1912$ |
| 27225.7-f2 |
27225.7-f |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.7 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{20} \cdot 5^{7} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$2.486362858$ |
$0.163764567$ |
7.857205002 |
\( -\frac{772786757876}{263063295} a + \frac{53647169159}{87687765} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -238 a - 2055\) , \( 2766 a + 41842\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-238a-2055\right){x}+2766a+41842$ |
*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.
