| 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 |
| 57.1-d3 |
57.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$36.19446804$ |
0.299629650 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -18372 a - 60161\) , \( 2300500 a + 7533946\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-18372a-60161\right){x}+2300500a+7533946$ |
| 57.1-g3 |
57.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.85372$ |
$(4a+13), (10a-43)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.892773922$ |
$18.84609635$ |
4.858622600 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2{x}-1$ |
| 171.1-e3 |
171.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{8} \cdot 19^{2} \) |
$2.43964$ |
$(4a+13), (10a-43)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$15.07895257$ |
0.998628029 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -182 a - 593\) , \( -2528 a - 8280\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-182a-593\right){x}-2528a-8280$ |
| 171.1-h3 |
171.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{8} \cdot 19^{2} \) |
$2.43964$ |
$(4a+13), (10a-43)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$15.07895257$ |
0.998628029 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 182 a - 775\) , \( 2528 a - 10808\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(182a-775\right){x}+2528a-10808$ |
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*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.
