| 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 |
| 48.5-c3 |
48.5-c |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
48.5 |
\( 2^{4} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$1.77577$ |
$(a+3), (4a+13)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$20.16176987$ |
1.335245828 |
\( -\frac{833}{9} a + \frac{20207}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -51 a - 167\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-51a-167\right){x}$ |
| 48.5-d3 |
48.5-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
48.5 |
\( 2^{4} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$1.77577$ |
$(a+3), (4a+13)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.482088370$ |
$22.24083115$ |
2.183019872 |
\( -\frac{833}{9} a + \frac{20207}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 96 a - 408\) , \( -423 a + 1809\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(96a-408\right){x}-423a+1809$ |
| 144.4-c3 |
144.4-c |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
144.4 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{10} \) |
$2.33704$ |
$(a+3), (4a+13)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.511546416$ |
$10.86445939$ |
4.350329747 |
\( -\frac{833}{9} a + \frac{20207}{9} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 7 a + 16\) , \( 7 a + 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a+16\right){x}+7a+21$ |
| 144.4-e3 |
144.4-e |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
144.4 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{10} \) |
$2.33704$ |
$(a+3), (4a+13)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.652070110$ |
$13.75784114$ |
2.376496351 |
\( -\frac{833}{9} a + \frac{20207}{9} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -47624 a - 155963\) , \( 2790926 a + 9140052\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-47624a-155963\right){x}+2790926a+9140052$ |
| 192.6-b3 |
192.6-b |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.6 |
\( 2^{6} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.51132$ |
$(a+3), (4a+13)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$13.30619092$ |
0.881224021 |
\( -\frac{833}{9} a + \frac{20207}{9} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 5 a - 31\) , \( 24 a - 108\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(5a-31\right){x}+24a-108$ |
| 192.6-q3 |
192.6-q |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.6 |
\( 2^{6} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.51132$ |
$(a+3), (4a+13)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.257879847$ |
$16.84984538$ |
5.614714104 |
\( -\frac{833}{9} a + \frac{20207}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2784 a - 9113\) , \( 37294 a + 122137\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2784a-9113\right){x}+37294a+122137$ |
*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.
