| 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 |
| 15.1-a5 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 382 a - 1057\) , \( -6554 a + 18298\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(382a-1057\right){x}-6554a+18298$ |
| 15.1-b5 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a - 84\) , \( -3 a - 310\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a-84\right){x}-3a-310$ |
| 45.2-a5 |
45.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{16} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$1.477614779$ |
1.289767919 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 224 a - 687\) , \( 3434 a - 9366\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(224a-687\right){x}+3434a-9366$ |
| 45.2-b5 |
45.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{16} \) |
$1.06060$ |
$(-a+2), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.283894188$ |
$2.245920702$ |
1.258473281 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -297 a - 579\) , \( -5300 a - 9353\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-297a-579\right){x}-5300a-9353$ |
| 1875.1-k5 |
1875.1-k |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
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 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 9480 a - 26571\) , \( -787207 a + 2197723\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(9480a-26571\right){x}-787207a+2197723$ |
| 1875.1-bj5 |
1875.1-bj |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
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 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -128 a - 2125\) , \( -2616 a - 39384\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-128a-2125\right){x}-2616a-39384$ |
*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.
