| 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 |
| 48.4-e3 |
48.4-e |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
48.4 |
\( 2^{4} \cdot 3 \) |
\( 2^{14} \cdot 3^{4} \) |
$1.77577$ |
$(a-4), (4a+13)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$18.49586432$ |
1.224918538 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -9704 a - 31777\) , \( 958322 a + 3138427\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-9704a-31777\right){x}+958322a+3138427$ |
| 48.4-f3 |
48.4-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
48.4 |
\( 2^{4} \cdot 3 \) |
\( 2^{14} \cdot 3^{4} \) |
$1.77577$ |
$(a-4), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.184394869$ |
$3.378573647$ |
1.060040542 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 347 a - 1483\) , \( 7257 a - 31023\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(347a-1483\right){x}+7257a-31023$ |
| 96.4-c3 |
96.4-c |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
96.4 |
\( 2^{5} \cdot 3 \) |
\( 2^{14} \cdot 3^{4} \) |
$2.11176$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$3.378573647$ |
3.580024093 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 64 a - 280\) , \( 553 a - 2375\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(64a-280\right){x}+553a-2375$ |
| 96.4-k3 |
96.4-k |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
96.4 |
\( 2^{5} \cdot 3 \) |
\( 2^{14} \cdot 3^{4} \) |
$2.11176$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$18.49586432$ |
2.449837077 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -52161 a - 170820\) , \( 11740716 a + 38449872\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-52161a-170820\right){x}+11740716a+38449872$ |
| 144.5-f3 |
144.5-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
144.5 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{10} \) |
$2.33704$ |
$(a-4), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$2.998456844$ |
3.177242489 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 314230 a - 1343303\) , \( 186465148 a - 797123070\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(314230a-1343303\right){x}+186465148a-797123070$ |
| 144.5-g3 |
144.5-g |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
144.5 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{10} \) |
$2.33704$ |
$(a-4), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.946866677$ |
1.840269938 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -90 a - 326\) , \( -1235 a - 3991\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-90a-326\right){x}-1235a-3991$ |
| 192.7-d3 |
192.7-d |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.7 |
\( 2^{6} \cdot 3 \) |
\( 2^{20} \cdot 3^{4} \) |
$2.51132$ |
$(a-4), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$8.508139335$ |
1.126930584 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -731 a - 2400\) , \( -21780 a - 71329\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-731a-2400\right){x}-21780a-71329$ |
| 192.7-l3 |
192.7-l |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.7 |
\( 2^{6} \cdot 3 \) |
\( 2^{20} \cdot 3^{4} \) |
$2.51132$ |
$(a-4), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.579639355$ |
$3.672344642$ |
3.073434351 |
\( -\frac{24913903427}{36} a + \frac{106505465453}{36} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 18355 a - 78466\) , \( 2660510 a - 11373460\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(18355a-78466\right){x}+2660510a-11373460$ |
*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.
