| 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 |
| 27225.3-a1 |
27225.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{6} \cdot 11^{8} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.697491699$ |
$0.572804391$ |
3.854774863 |
\( -\frac{393194}{11} a - 1506561 \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 113 a - 530\) , \( 1268 a - 4412\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(113a-530\right){x}+1268a-4412$ |
| 27225.3-a2 |
27225.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{6} \cdot 11^{7} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.174372924$ |
$1.145608783$ |
3.854774863 |
\( -\frac{7136}{11} a + \frac{11895}{11} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 3 a - 35\) , \( 25 a - 56\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a-35\right){x}+25a-56$ |
| 27225.3-b1 |
27225.3-b |
$2$ |
$11$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{10} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$11$ |
11B.10.4[2] |
|
\( 2 \) |
$1$ |
$0.219357806$ |
5.612608756 |
\( -24729001 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 2311 a + 1983\) , \( -19721 a + 138914\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(2311a+1983\right){x}-19721a+138914$ |
| 27225.3-b2 |
27225.3-b |
$2$ |
$11$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{2} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$11$ |
11B.10.2[2] |
|
\( 2 \) |
$1$ |
$2.412935876$ |
5.612608756 |
\( -121 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( a + 3\) , \( a - 10\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+3\right){x}+a-10$ |
| 27225.3-c1 |
27225.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 5^{2} \cdot 11^{10} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.692633907$ |
$0.748828780$ |
2.502130266 |
\( \frac{94706}{363} a + \frac{524143}{363} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 28 a - 81\) , \( -19 a - 172\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(28a-81\right){x}-19a-172$ |
| 27225.3-d1 |
27225.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 5^{8} \cdot 11^{10} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$0.334886411$ |
1.615552834 |
\( \frac{94706}{363} a + \frac{524143}{363} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 186 a + 180\) , \( 338 a + 1367\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(186a+180\right){x}+338a+1367$ |
| 27225.3-e1 |
27225.3-e |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{8} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$47.86298431$ |
$0.028828495$ |
6.656491569 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 602166 a + 516141\) , \( -81612670 a + 587936165\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(602166a+516141\right){x}-81612670a+587936165$ |
| 27225.3-e2 |
27225.3-e |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{16} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5Cs.4.1 |
$1$ |
\( 2^{2} \) |
$9.572596863$ |
$0.144142475$ |
6.656491569 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 796 a + 681\) , \( -6640 a + 50405\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(796a+681\right){x}-6640a+50405$ |
| 27225.3-e3 |
27225.3-e |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{6} \cdot 11^{8} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$1.914519372$ |
$0.720712377$ |
6.656491569 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 26 a + 21\) , \( 70 a - 415\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(26a+21\right){x}+70a-415$ |
| 27225.3-f1 |
27225.3-f |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{20} \cdot 5^{7} \cdot 11^{8} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$2.486362858$ |
$0.163764567$ |
7.857205002 |
\( \frac{772786757876}{263063295} a - \frac{611845250399}{263063295} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 236 a - 2292\) , \( -2767 a + 44609\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(236a-2292\right){x}-2767a+44609$ |
| 27225.3-f2 |
27225.3-f |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.3 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{13} \cdot 5^{8} \cdot 11^{7} \) |
$3.80695$ |
$(-a), (a-2), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$4.972725717$ |
$0.327529135$ |
7.857205002 |
\( -\frac{2252642504}{601425} a + \frac{109962161}{601425} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 291 a - 477\) , \( 3338 a - 1426\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(291a-477\right){x}+3338a-1426$ |
*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.
