| 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 |
| 16.3-a1 |
16.3-a |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{18} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$1$ |
$214.2397030$ |
1.113899608 |
\( \frac{234102825}{8} a^{2} - \frac{51209311}{4} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + a + 4\) , \( a + 1\) , \( -10 a^{3} - 24 a^{2} - 3 a + 8\) , \( 47 a^{3} + 92 a^{2} - 47 a - 63\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-10a^{3}-24a^{2}-3a+8\right){x}+47a^{3}+92a^{2}-47a-63$ |
| 16.3-a2 |
16.3-a |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{24} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$107.1198515$ |
1.113899608 |
\( -\frac{159367}{64} a^{2} + \frac{35185}{32} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} - 4 a + 1\) , \( -119 a^{3} - 79 a^{2} + 544 a + 365\) , \( 17497 a^{3} + 11587 a^{2} - 79812 a - 52851\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(-119a^{3}-79a^{2}+544a+365\right){x}+17497a^{3}+11587a^{2}-79812a-52851$ |
| 16.3-b1 |
16.3-b |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.096670278$ |
$1126.453841$ |
2.264709247 |
\( 644 a^{2} - 1528 \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a + 1\) , \( 47 a^{3} + 28 a^{2} - 216 a - 130\) , \( -444 a^{3} - 296 a^{2} + 2025 a + 1349\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(47a^{3}+28a^{2}-216a-130\right){x}-444a^{3}-296a^{2}+2025a+1349$ |
| 16.3-b2 |
16.3-b |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{10} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.048335139$ |
$2252.907682$ |
2.264709247 |
\( -867342 a^{2} + 3958532 \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{3} - 4 a + 1\) , \( -3 a^{3} - 7 a^{2} + 2 a + 7\) , \( -5 a^{3} - 9 a^{2} + 5 a + 3\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(-3a^{3}-7a^{2}+2a+7\right){x}-5a^{3}-9a^{2}+5a+3$ |
| 16.3-c1 |
16.3-c |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.096670278$ |
$1126.453841$ |
2.264709247 |
\( 644 a^{2} - 1528 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} - a + 3\) , \( a^{3} - 4 a + 1\) , \( 2 a^{3} - a^{2} - 9 a + 5\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(2a^{3}-a^{2}-9a+5\right){x}$ |
| 16.3-c2 |
16.3-c |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{10} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.048335139$ |
$2252.907682$ |
2.264709247 |
\( -867342 a^{2} + 3958532 \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( a^{3} - 4 a + 1\) , \( 5 a^{3} - 5 a^{2} + 5\) , \( -5 a^{3} + 21 a^{2} + 2 a - 11\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(5a^{3}-5a^{2}+5\right){x}-5a^{3}+21a^{2}+2a-11$ |
| 16.3-d1 |
16.3-d |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{18} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$1$ |
$214.2397030$ |
1.113899608 |
\( \frac{234102825}{8} a^{2} - \frac{51209311}{4} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} - a^{2} + 4 a + 4\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 8 a^{3} - 26 a^{2} + 12 a + 8\) , \( -71 a^{3} + 145 a^{2} + 51 a - 83\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+4\right){x}^{2}+\left(8a^{3}-26a^{2}+12a+8\right){x}-71a^{3}+145a^{2}+51a-83$ |
| 16.3-d2 |
16.3-d |
$2$ |
$2$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{24} \) |
$12.15283$ |
$(-a^3+4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$107.1198515$ |
1.113899608 |
\( -\frac{159367}{64} a^{2} + \frac{35185}{32} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{2} - 3\) , \( 117 a^{3} - 78 a^{2} - 532 a + 361\) , \( -17215 a^{3} + 11402 a^{2} + 78529 a - 52007\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(117a^{3}-78a^{2}-532a+361\right){x}-17215a^{3}+11402a^{2}+78529a-52007$ |
*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.
