| 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-a1 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -379 a - 681\) , \( 5873 a + 10526\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-379a-681\right){x}+5873a+10526$ |
| 15.2-b1 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 5 a - 90\) , \( 8 a - 403\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(5a-90\right){x}+8a-403$ |
| 45.1-a1 |
45.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{16} \) |
$1.06060$ |
$(-a+2), (-a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$1.477614779$ |
1.289767919 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -224 a - 463\) , \( -3434 a - 5932\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-224a-463\right){x}-3434a-5932$ |
| 45.1-b1 |
45.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{16} \) |
$1.06060$ |
$(-a+2), (-a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.283894188$ |
$2.245920702$ |
1.258473281 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 302 a - 883\) , \( 4418 a - 12272\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(302a-883\right){x}+4418a-12272$ |
| 1875.1-e1 |
1875.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1875.1 |
\( 3 \cdot 5^{4} \) |
\( - 3 \cdot 5^{28} \) |
$2.69463$ |
$(-a+2), (-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$0.244129907$ |
1.704752426 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 126 a - 2252\) , \( 2615 a - 41999\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(126a-2252\right){x}+2615a-41999$ |
| 1875.1-bp1 |
1875.1-bp |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1875.1 |
\( 3 \cdot 5^{4} \) |
\( - 3 \cdot 5^{28} \) |
$2.69463$ |
$(-a+2), (-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.880738313$ |
$1.631232646$ |
5.525614808 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -9481 a - 17090\) , \( 787207 a + 1410516\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-9481a-17090\right){x}+787207a+1410516$ |
*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.
