| 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-e2 |
48.5-e |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
48.5 |
\( 2^{4} \cdot 3 \) |
\( - 2^{20} \cdot 3 \) |
$1.77577$ |
$(a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$9.247932164$ |
1.224918538 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -1353 a - 4428\) , \( -26659 a - 87307\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1353a-4428\right){x}-26659a-87307$ |
| 48.5-f2 |
48.5-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
48.5 |
\( 2^{4} \cdot 3 \) |
\( - 2^{20} \cdot 3 \) |
$1.77577$ |
$(a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.296098717$ |
$13.51429459$ |
1.060040542 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -8 a + 32\) , \( -34 a + 146\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-8a+32\right){x}-34a+146$ |
| 96.3-c2 |
96.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
96.3 |
\( 2^{5} \cdot 3 \) |
\( - 2^{20} \cdot 3 \) |
$2.11176$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$13.51429459$ |
3.580024093 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -43 a + 185\) , \( -470 a + 2011\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-43a+185\right){x}-470a+2011$ |
| 96.3-k2 |
96.3-k |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
96.3 |
\( 2^{5} \cdot 3 \) |
\( - 2^{20} \cdot 3 \) |
$2.11176$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$9.247932164$ |
2.449837077 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -254 a - 829\) , \( -1709 a - 5595\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-254a-829\right){x}-1709a-5595$ |
| 144.4-f2 |
144.4-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
144.4 |
\( 2^{4} \cdot 3^{2} \) |
\( - 2^{20} \cdot 3^{7} \) |
$2.33704$ |
$(a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$5.996913689$ |
3.177242489 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -14 a - 41\) , \( 14 a + 48\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-14a-41\right){x}+14a+48$ |
| 144.4-g2 |
144.4-g |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
144.4 |
\( 2^{4} \cdot 3^{2} \) |
\( - 2^{20} \cdot 3^{7} \) |
$2.33704$ |
$(a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.946866677$ |
1.840269938 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -7286 a + 31183\) , \( -910322 a + 3891612\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-7286a+31183\right){x}-910322a+3891612$ |
| 192.6-d2 |
192.6-d |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.6 |
\( 2^{6} \cdot 3 \) |
\( - 2^{26} \cdot 3 \) |
$2.51132$ |
$(a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$8.508139335$ |
1.126930584 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -71829 a - 235234\) , \( -8626826 a - 28252141\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-71829a-235234\right){x}-8626826a-28252141$ |
| 192.6-l2 |
192.6-l |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.6 |
\( 2^{6} \cdot 3 \) |
\( - 2^{26} \cdot 3 \) |
$2.51132$ |
$(a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.579639355$ |
$7.344689285$ |
3.073434351 |
\( \frac{152551}{768} a + \frac{476915}{384} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -2 a + 5\) , \( -a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2a+5\right){x}-a+2$ |
*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.
