| 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 |
| 10.2-a1 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$4.047893897$ |
1.081845150 |
\( -\frac{18258829169}{8000} a + \frac{68318145777}{8000} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -18 a - 68\) , \( -814 a - 3046\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-18a-68\right){x}-814a-3046$ |
| 10.2-a2 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$36.43104508$ |
1.081845150 |
\( -\frac{5689}{20} a + \frac{21627}{20} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 2 a + 7\) , \( 30 a + 112\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(2a+7\right){x}+30a+112$ |
| 10.2-b1 |
10.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5 \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$0.331361056$ |
2.213999186 |
\( -\frac{7114676554418062503}{10} a + \frac{26620682081199569989}{10} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 17221 a - 64425\) , \( 2362884 a - 8841125\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(17221a-64425\right){x}+2362884a-8841125$ |
| 10.2-b2 |
10.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{10} \cdot 5^{5} \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$8.284026412$ |
2.213999186 |
\( -\frac{73603923}{100000} a + \frac{358833109}{100000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 31 a - 95\) , \( 114 a - 405\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(31a-95\right){x}+114a-405$ |
| 10.2-c1 |
10.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5 \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$0.272002473$ |
$14.16313473$ |
2.059198521 |
\( -\frac{7114676554418062503}{10} a + \frac{26620682081199569989}{10} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -1285 a - 4941\) , \( 48650 a + 182431\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-1285a-4941\right){x}+48650a+182431$ |
| 10.2-c2 |
10.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{10} \cdot 5^{5} \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$0.054400494$ |
$14.16313473$ |
2.059198521 |
\( -\frac{73603923}{100000} a + \frac{358833109}{100000} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 5 a + 29\) , \( 20 a + 81\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(5a+29\right){x}+20a+81$ |
| 10.2-d1 |
10.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.038349955$ |
$20.19548844$ |
1.241956719 |
\( -\frac{18258829169}{8000} a + \frac{68318145777}{8000} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 20 a - 72\) , \( -128 a + 480\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(20a-72\right){x}-128a+480$ |
| 10.2-d2 |
10.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$1.18914$ |
$(-a+4), (-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.115049867$ |
$20.19548844$ |
1.241956719 |
\( -\frac{5689}{20} a + \frac{21627}{20} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+3{x}+1$ |
*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.
