| 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 |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{421}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$2.59295$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$55.92996476$ |
2.725859692 |
\( \frac{23529}{2} a + \frac{233401}{2} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -435 a - 4254\) , \( 10578 a + 103244\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-435a-4254\right){x}+10578a+103244$ |
| 4.1-b1 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{421}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$2.59295$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs |
$9$ |
\( 1 \) |
$1$ |
$13.16799490$ |
5.775919227 |
\( \frac{521660125}{32768} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 161813422578 a - 1740973633375\) , \( -107148472674788715 a + 1152825660914971497\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(161813422578a-1740973633375\right){x}-107148472674788715a+1152825660914971497$ |
| 4.1-b2 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{421}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$2.59295$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$13.16799490$ |
5.775919227 |
\( -\frac{5436766619625}{32} a + \frac{29247476296625}{16} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 284883 a - 3064675\) , \( -263169516 a + 2831479929\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(284883a-3064675\right){x}-263169516a+2831479929$ |
| 4.1-b3 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{421}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$2.59295$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$13.16799490$ |
5.775919227 |
\( \frac{5436766619625}{32} a + \frac{53058185973625}{32} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -284882 a - 2779898\) , \( 263454398 a + 2571090206\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-284882a-2779898\right){x}+263454398a+2571090206$ |
| 4.1-c1 |
4.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{421}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$2.59295$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$55.92996476$ |
2.725859692 |
\( -\frac{23529}{2} a + 128465 \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 486 a - 4793\) , \( -15320 a + 166942\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(486a-4793\right){x}-15320a+166942$ |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{421}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{5} \) |
$2.74171$ |
$(a+10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$4.007185211$ |
$3.190522549$ |
6.231034938 |
\( -\frac{795683851}{3125} a - \frac{7766066319}{3125} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 8 a + 193\) , \( 80 a + 36\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(8a+193\right){x}+80a+36$ |
| 5.2-a1 |
5.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{421}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{5} \) |
$2.74171$ |
$(-a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$4.007185211$ |
$3.190522549$ |
6.231034938 |
\( \frac{795683851}{3125} a - \frac{1712350034}{625} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -8 a + 201\) , \( -80 a + 116\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-8a+201\right){x}-80a+116$ |
*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.
