| 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 |
| 3.1-a1 |
3.1-a |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1255.113391$ |
2.484364996 |
\( -\frac{4151552}{27} a^{3} + \frac{8435392}{27} a^{2} + \frac{6494080}{27} a - \frac{372736}{27} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - a^{2} - 6 a + 1\) , \( a^{2} - 2\) , \( 3 a^{3} - 10 a^{2} - 41 a - 12\) , \( -3 a^{3} - 35 a^{2} - 74 a - 27\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+1\right){x}^{2}+\left(3a^{3}-10a^{2}-41a-12\right){x}-3a^{3}-35a^{2}-74a-27$ |
| 3.1-a2 |
3.1-a |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1255.113391$ |
2.484364996 |
\( \frac{1460648042528}{27} a^{3} - \frac{931716587896}{27} a^{2} - \frac{8169571713064}{27} a + \frac{2289903595912}{27} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( a^{3} - 7 a - 1\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( -33 a^{3} + 77 a^{2} + 25 a - 11\) , \( -292 a^{3} + 645 a^{2} + 324 a - 132\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(a^{3}-7a-1\right){x}^{2}+\left(-33a^{3}+77a^{2}+25a-11\right){x}-292a^{3}+645a^{2}+324a-132$ |
| 3.1-a3 |
3.1-a |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$2510.226783$ |
2.484364996 |
\( \frac{78693248}{729} a^{3} - \frac{50155840}{729} a^{2} - \frac{440181376}{729} a + \frac{124901248}{729} \) |
\( \bigl[a^{3} - 5 a\) , \( -2 a^{3} + a^{2} + 10 a - 2\) , \( 1\) , \( -13 a^{3} + 8 a^{2} + 72 a - 19\) , \( 69 a^{3} - 44 a^{2} - 386 a + 108\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+{y}={x}^{3}+\left(-2a^{3}+a^{2}+10a-2\right){x}^{2}+\left(-13a^{3}+8a^{2}+72a-19\right){x}+69a^{3}-44a^{2}-386a+108$ |
| 3.1-a4 |
3.1-a |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{12} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$627.5566957$ |
2.484364996 |
\( \frac{29828142752}{531441} a^{3} + \frac{82452366632}{531441} a^{2} + \frac{20522751224}{531441} a - \frac{11772688328}{531441} \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( a^{3} - 8 a^{2} + 9 a + 15\) , \( a^{3} - 12 a^{2} + 19 a + 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(a^{3}-8a^{2}+9a+15\right){x}+a^{3}-12a^{2}+19a+15$ |
| 3.1-b1 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.226328221$ |
$255.3488237$ |
1.372733573 |
\( -\frac{4151552}{27} a^{3} + \frac{8435392}{27} a^{2} + \frac{6494080}{27} a - \frac{372736}{27} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 6 a + 1\) , \( 1\) , \( -7 a^{3} + 3 a^{2} + 40 a - 3\) , \( -35 a^{3} + 22 a^{2} + 196 a - 53\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+6a+1\right){x}^{2}+\left(-7a^{3}+3a^{2}+40a-3\right){x}-35a^{3}+22a^{2}+196a-53$ |
| 3.1-b2 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 3 \) |
$0.113164110$ |
$1021.395294$ |
1.372733573 |
\( \frac{78693248}{729} a^{3} - \frac{50155840}{729} a^{2} - \frac{440181376}{729} a + \frac{124901248}{729} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - 6 a\) , \( 26 a^{3} + 7 a^{2} - 158 a - 91\) , \( -10 a^{3} - 4 a^{2} + 61 a + 35\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(26a^{3}+7a^{2}-158a-91\right){x}-10a^{3}-4a^{2}+61a+35$ |
| 3.1-b3 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.226328221$ |
$255.3488237$ |
1.372733573 |
\( \frac{1460648042528}{27} a^{3} - \frac{931716587896}{27} a^{2} - \frac{8169571713064}{27} a + \frac{2289903595912}{27} \) |
\( \bigl[-a^{3} + a^{2} + 6 a - 2\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a\) , \( 7 a^{3} - 3 a^{2} - 39 a + 7\) , \( -a^{3} + a^{2} + 8 a\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+6a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(7a^{3}-3a^{2}-39a+7\right){x}-a^{3}+a^{2}+8a$ |
| 3.1-b4 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{12} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.056582055$ |
$1021.395294$ |
1.372733573 |
\( \frac{29828142752}{531441} a^{3} + \frac{82452366632}{531441} a^{2} + \frac{20522751224}{531441} a - \frac{11772688328}{531441} \) |
\( \bigl[a + 1\) , \( -2 a^{3} + a^{2} + 12 a - 2\) , \( a\) , \( 4 a^{3} - 6 a^{2} - 28 a + 10\) , \( -4 a^{3} + 5 a^{2} + 28 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-2a^{3}+a^{2}+12a-2\right){x}^{2}+\left(4a^{3}-6a^{2}-28a+10\right){x}-4a^{3}+5a^{2}+28a-7$ |
*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.
