| 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 |
| 868.2-b1 |
868.2-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
868.2 |
\( 2^{2} \cdot 7 \cdot 31 \) |
\( 2^{18} \cdot 7^{2} \cdot 31 \) |
$0.84010$ |
$(-3a+1), (6a-5), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.497046915$ |
1.147880680 |
\( -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -1457 a - 824\) , \( -35350 a - 779\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1457a-824\right){x}-35350a-779$ |
| 54684.2-a5 |
54684.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{18} \cdot 3^{6} \cdot 7^{8} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$9$ |
\( 2 \) |
$1$ |
$0.108464529$ |
2.254392903 |
\( -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -32887 a + 47339\) , \( 1679851 a + 2156108\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-32887a+47339\right){x}+1679851a+2156108$ |
| 55552.2-e5 |
55552.2-e |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
55552.2 |
\( 2^{8} \cdot 7 \cdot 31 \) |
\( 2^{42} \cdot 7^{2} \cdot 31 \) |
$2.37615$ |
$(-3a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \) |
$2.657384984$ |
$0.124261728$ |
3.050360885 |
\( -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -13179 a + 36502\) , \( 2179225 a + 46814\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-13179a+36502\right){x}+2179225a+46814$ |
| 146692.2-b5 |
146692.2-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{18} \cdot 7^{2} \cdot 13^{6} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$9$ |
\( 2^{2} \) |
$0.252641283$ |
$0.137856010$ |
2.895555473 |
\( -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 3614 a + 24016\) , \( -1825903 a + 1200467\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3614a+24016\right){x}-1825903a+1200467$ |
| 146692.6-b5 |
146692.6-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.6 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{18} \cdot 7^{2} \cdot 13^{6} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$4.988945552$ |
$0.137856010$ |
3.176609500 |
\( -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -22560 a + 28455\) , \( -532657 a - 1274643\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-22560a+28455\right){x}-532657a-1274643$ |
*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.
