| 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 |
| 8.2-a1 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.348459975$ |
3.828605526 |
\( -\frac{1157415}{1024} a - \frac{5901817}{1024} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -284341 a + 1756911\) , \( 837230820 a - 5173172497\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-284341a+1756911\right){x}+837230820a-5173172497$ |
| 8.2-a2 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{38} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$4.348459975$ |
3.828605526 |
\( \frac{172032790985}{1073741824} a + \frac{790355237143}{1073741824} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 2580394 a - 15944024\) , \( -23701425878 a + 146448938169\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2580394a-15944024\right){x}-23701425878a+146448938169$ |
| 8.2-b1 |
8.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$6.235527253$ |
$1.771317458$ |
1.944933359 |
\( -\frac{47232606203}{8} a - \frac{244609748133}{8} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 17379 a - 107319\) , \( 2996015 a - 18512003\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(17379a-107319\right){x}+2996015a-18512003$ |
| 8.2-b2 |
8.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{9} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.078509084$ |
$15.94185712$ |
1.944933359 |
\( -\frac{2307}{2} a + \frac{7523}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 974 a - 5954\) , \( -33754 a + 208662\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(974a-5954\right){x}-33754a+208662$ |
| 8.2-c1 |
8.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$29.10944917$ |
2.562944090 |
\( -\frac{47232606203}{8} a - \frac{244609748133}{8} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -869257 a - 4501774\) , \( 1049040547 a + 5432884883\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-869257a-4501774\right){x}+1049040547a+5432884883$ |
| 8.2-c2 |
8.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{9} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$29.10944917$ |
2.562944090 |
\( -\frac{2307}{2} a + \frac{7523}{2} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -10102 a - 52289\) , \( 1593581 a + 8253049\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10102a-52289\right){x}+1593581a+8253049$ |
| 8.2-d1 |
8.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.637207042$ |
$19.70056279$ |
1.473679633 |
\( -\frac{1157415}{1024} a - \frac{5901817}{1024} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -93 a - 460\) , \( 979 a + 5098\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-93a-460\right){x}+979a+5098$ |
| 8.2-d2 |
8.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{38} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.911621126$ |
$2.188951422$ |
1.473679633 |
\( \frac{172032790985}{1073741824} a + \frac{790355237143}{1073741824} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 522 a + 2725\) , \( 1695 a + 8806\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(522a+2725\right){x}+1695a+8806$ |
*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.
