| 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 |
| 16.1-a1 |
16.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$1.55803$ |
$(-3a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$9.924776877$ |
2.276899970 |
\( -\frac{27}{8} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 6 a + 26\) , \( 8 a + 36\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a+26\right){x}+8a+36$ |
| 16.1-c1 |
16.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$1.55803$ |
$(-3a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$9.924776877$ |
2.276899970 |
\( -\frac{27}{8} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 5 a + 16\) , \( 8 a + 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a+16\right){x}+8a+31$ |
| 18.2-a1 |
18.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.303728707$ |
4.338523642 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -a - 2\) , \( -9 a - 44\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-a-2\right){x}-9a-44$ |
| 18.2-b1 |
18.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.123958993$ |
$20.83448291$ |
1.184988031 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -a - 7\) , \( 8 a + 34\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a-7\right){x}+8a+34$ |
| 18.3-a1 |
18.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.3 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.303728707$ |
4.338523642 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -2\) , \( 9 a - 44\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}-2{x}+9a-44$ |
| 18.3-b1 |
18.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.3 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.123958993$ |
$20.83448291$ |
1.184988031 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -7\) , \( -9 a + 34\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-7{x}-9a+34$ |
| 50.2-a1 |
50.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
50.2 |
\( 2 \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{6} \) |
$2.07151$ |
$(-3a+13), (-2a+9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \) |
$1$ |
$8.876990303$ |
4.073042490 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -66 a - 283\) , \( -13884 a - 60517\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-66a-283\right){x}-13884a-60517$ |
| 50.2-n1 |
50.2-n |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
50.2 |
\( 2 \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{6} \) |
$2.07151$ |
$(-3a+13), (-2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.059892692$ |
$8.876990303$ |
1.463672891 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -66 a - 288\) , \( 13818 a + 60231\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-66a-288\right){x}+13818a+60231$ |
| 50.3-a1 |
50.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{6} \) |
$2.07151$ |
$(-3a+13), (2a+9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \) |
$1$ |
$8.876990303$ |
4.073042490 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 66 a - 283\) , \( 13884 a - 60517\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(66a-283\right){x}+13884a-60517$ |
| 50.3-n1 |
50.3-n |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{6} \) |
$2.07151$ |
$(-3a+13), (2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.059892692$ |
$8.876990303$ |
1.463672891 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 66 a - 288\) , \( -13818 a + 60231\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(66a-288\right){x}-13818a+60231$ |
*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.
