| 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 |
| 19.1-a1 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$0.81321$ |
$(-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$0.935309008$ |
0.858298410 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -769\) , \( -8470\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-769{x}-8470$ |
| 475.1-a1 |
475.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
475.1 |
\( 5^{2} \cdot 19 \) |
\( 5^{6} \cdot 19^{2} \) |
$1.81840$ |
$(-a), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cn, 3B |
$9$ |
\( 2 \) |
$1$ |
$0.418282904$ |
3.454584462 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -769 a + 3845\) , \( -33879 a - 42349\bigr] \) |
${y}^2+{y}={x}^3-a{x}^2+\left(-769a+3845\right){x}-33879a-42349$ |
| 475.3-a1 |
475.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
475.3 |
\( 5^{2} \cdot 19 \) |
\( 5^{6} \cdot 19^{2} \) |
$1.81840$ |
$(a-1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cn, 3B |
$9$ |
\( 2 \) |
$1$ |
$0.418282904$ |
3.454584462 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 769 a + 3076\) , \( 33879 a - 76228\bigr] \) |
${y}^2+{y}={x}^3+\left(a-1\right){x}^2+\left(769a+3076\right){x}+33879a-76228$ |
| 1539.1-a1 |
1539.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
1539.1 |
\( 3^{4} \cdot 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.43964$ |
$(-2a+1), (3)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$4.678741769$ |
$0.311769669$ |
2.379707628 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -6924\) , \( 221760\bigr] \) |
${y}^2+{y}={x}^3-6924{x}+221760$ |
| 4864.1-f1 |
4864.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
4864.1 |
\( 2^{8} \cdot 19 \) |
\( 2^{24} \cdot 19^{2} \) |
$3.25285$ |
$(-2a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2Cn, 3B |
$1$ |
\( 2 \) |
$16.54388095$ |
$0.233827252$ |
7.099793362 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -12309\) , \( 529757\bigr] \) |
${y}^2={x}^3-{x}^2-12309{x}+529757$ |
| 5491.1-a1 |
5491.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
5491.1 |
\( 17^{2} \cdot 19 \) |
\( 17^{6} \cdot 19^{2} \) |
$3.35296$ |
$(a+3), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cn, 3B |
$1$ |
\( 2 \) |
$8.069824998$ |
$0.226845754$ |
3.359757714 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -5385 a - 3079\) , \( -265648 a + 197882\bigr] \) |
${y}^2+{y}={x}^3+a{x}^2+\left(-5385a-3079\right){x}-265648a+197882$ |
| 5491.3-a1 |
5491.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
5491.3 |
\( 17^{2} \cdot 19 \) |
\( 17^{6} \cdot 19^{2} \) |
$3.35296$ |
$(a-4), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cn, 3B |
$1$ |
\( 2 \) |
$8.069824998$ |
$0.226845754$ |
3.359757714 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 5385 a - 8464\) , \( 265648 a - 67766\bigr] \) |
${y}^2+{y}={x}^3+\left(-a+1\right){x}^2+\left(5385a-8464\right){x}+265648a-67766$ |
| 11875.3-a1 |
11875.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
11875.3 |
\( 5^{4} \cdot 19 \) |
\( 5^{12} \cdot 19^{2} \) |
$4.06606$ |
$(-a), (a-1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2Cn, 3B |
$9$ |
\( 2 \) |
$1$ |
$0.187061801$ |
1.544937138 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -19233\) , \( -1020257\bigr] \) |
${y}^2+{y}={x}^3-{x}^2-19233{x}-1020257$ |
*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.
