| 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 |
| 54.1-c1 |
54.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
54.1 |
\( 2 \cdot 3^{3} \) |
\( - 2^{3} \cdot 3^{18} \) |
$2.11176$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.845105407$ |
$3.079678287$ |
1.194178992 |
\( -\frac{672757314475}{236196} a + \frac{2932541135213}{236196} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 1402 a - 6104\) , \( 57726 a - 251618\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(1402a-6104\right){x}+57726a-251618$ |
| 54.1-e1 |
54.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
54.1 |
\( 2 \cdot 3^{3} \) |
\( - 2^{3} \cdot 3^{18} \) |
$2.11176$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \cdot 5 \) |
$0.100272282$ |
$5.939032190$ |
4.098651894 |
\( -\frac{672757314475}{236196} a + \frac{2932541135213}{236196} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 1398 a - 6093\) , \( -54926 a + 239417\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1398a-6093\right){x}-54926a+239417$ |
| 54.2-b1 |
54.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
54.2 |
\( 2 \cdot 3^{3} \) |
\( - 2^{3} \cdot 3^{18} \) |
$2.11176$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$5.939032190$ |
4.087522285 |
\( -\frac{672757314475}{236196} a + \frac{2932541135213}{236196} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 270340 a + 1178394\) , \( -385206082 a - 1679074385\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(270340a+1178394\right){x}-385206082a-1679074385$ |
| 54.2-f1 |
54.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
54.2 |
\( 2 \cdot 3^{3} \) |
\( - 2^{3} \cdot 3^{18} \) |
$2.11176$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$3.079678287$ |
2.826106617 |
\( -\frac{672757314475}{236196} a + \frac{2932541135213}{236196} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 270344 a + 1178405\) , \( 385746767 a + 1681431175\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(270344a+1178405\right){x}+385746767a+1681431175$ |
| 144.1-m1 |
144.1-m |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
144.1 |
\( 2^{4} \cdot 3^{2} \) |
\( - 2^{15} \cdot 3^{12} \) |
$2.69858$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$7.227848628$ |
1.658182197 |
\( -\frac{672757314475}{236196} a + \frac{2932541135213}{236196} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 43536 a - 189739\) , \( -10343971 a + 45088374\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(43536a-189739\right){x}-10343971a+45088374$ |
| 144.1-p1 |
144.1-p |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
144.1 |
\( 2^{4} \cdot 3^{2} \) |
\( - 2^{15} \cdot 3^{12} \) |
$2.69858$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$1.897899647$ |
4.354080404 |
\( -\frac{672757314475}{236196} a + \frac{2932541135213}{236196} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 43535 a - 189742\) , \( 10284826 a - 44830481\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(43535a-189742\right){x}+10284826a-44830481$ |
*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.
