| 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 |
| 225.3-CMa1 |
225.3-CMa |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.14784$ |
$(-a), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{yes}$ |
$-11$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$2.977452843$ |
1.795471620 |
\( -32768 \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 5 a + 3\) , \( a + 8\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(5a+3\right){x}+a+8$ |
| 225.3-a1 |
225.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{11} \cdot 5^{8} \) |
$1.14784$ |
$(-a), (a-2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.395636541$ |
$0.692871952$ |
1.554990352 |
\( -\frac{10449954757}{243} a - \frac{592340771}{243} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -330 a + 304\) , \( -1064 a + 5768\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-330a+304\right){x}-1064a+5768$ |
| 225.3-a2 |
225.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{21} \cdot 5^{8} \) |
$1.14784$ |
$(-a), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \) |
$0.465212180$ |
$0.692871952$ |
1.554990352 |
\( -\frac{507968593}{14348907} a + \frac{10568844616}{14348907} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -28 a + 75\) , \( -128 a + 243\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-28a+75\right){x}-128a+243$ |
| 225.3-a3 |
225.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{9} \cdot 5^{4} \) |
$1.14784$ |
$(-a), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \) |
$0.093042436$ |
$3.464359763$ |
1.554990352 |
\( \frac{24167}{27} a + \frac{28561}{27} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -3 a\) , \( a\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-3a{x}+a$ |
| 225.3-a4 |
225.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{7} \cdot 5^{4} \) |
$1.14784$ |
$(-a), (a-2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.279127308$ |
$3.464359763$ |
1.554990352 |
\( \frac{77363}{3} a + \frac{24904}{3} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -5 a + 4\) , \( 2 a - 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a+4\right){x}+2a-4$ |
| 225.3-b1 |
225.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{11} \cdot 5^{2} \) |
$1.14784$ |
$(-a), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1$ |
$1.549308785$ |
1.868536700 |
\( -\frac{10449954757}{243} a - \frac{592340771}{243} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 23 a + 93\) , \( 272 a - 400\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(23a+93\right){x}+272a-400$ |
| 225.3-b2 |
225.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{21} \cdot 5^{2} \) |
$1.14784$ |
$(-a), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1$ |
$1.549308785$ |
1.868536700 |
\( -\frac{507968593}{14348907} a + \frac{10568844616}{14348907} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 6 a + 3\) , \( 18 a - 23\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6a+3\right){x}+18a-23$ |
| 225.3-b3 |
225.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{9} \cdot 5^{10} \) |
$1.14784$ |
$(-a), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1$ |
$1.549308785$ |
1.868536700 |
\( \frac{24167}{27} a + \frac{28561}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 8 a - 20\) , \( -18 a + 25\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(8a-20\right){x}-18a+25$ |
| 225.3-b4 |
225.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{7} \cdot 5^{10} \) |
$1.14784$ |
$(-a), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1$ |
$1.549308785$ |
1.868536700 |
\( \frac{77363}{3} a + \frac{24904}{3} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -13 a - 27\) , \( 44 a + 50\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-13a-27\right){x}+44a+50$ |
*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.
