| 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 |
| 15.2-a6 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -175 a + 435\) , \( -146 a + 520\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-175a+435\right){x}-146a+520$ |
| 15.2-b6 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -275 a - 460\) , \( -4022 a - 7205\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-275a-460\right){x}-4022a-7205$ |
| 45.1-a6 |
45.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{4} \) |
$1.06060$ |
$(-a+2), (-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.738807389$ |
1.289767919 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -2342 a + 6520\) , \( -16356 a + 45512\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2342a+6520\right){x}-16356a+45512$ |
| 45.1-b6 |
45.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{4} \) |
$1.06060$ |
$(-a+2), (-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.320973547$ |
$4.491841405$ |
1.258473281 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -268 a - 13\) , \( 1712 a + 2212\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-268a-13\right){x}+1712a+2212$ |
| 1875.1-e6 |
1875.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1875.1 |
\( 3 \cdot 5^{4} \) |
\( - 3 \cdot 5^{16} \) |
$2.69463$ |
$(-a+2), (-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$0.244129907$ |
1.704752426 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -6874 a - 11502\) , \( -456885 a - 820249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-6874a-11502\right){x}-456885a-820249$ |
| 1875.1-bp6 |
1875.1-bp |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1875.1 |
\( 3 \cdot 5^{4} \) |
\( - 3 \cdot 5^{16} \) |
$2.69463$ |
$(-a+2), (-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.880738313$ |
$1.631232646$ |
5.525614808 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -4358 a + 10830\) , \( -35544 a + 119597\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4358a+10830\right){x}-35544a+119597$ |
*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.
