| 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 |
| 588.3-a1 |
588.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
588.3 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{11} \cdot 7^{2} \) |
$2.52778$ |
$(-a+3), (-2a+7), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \) |
$1$ |
$4.075245702$ |
2.128227658 |
\( \frac{1904464}{5103} a - \frac{7326080}{5103} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 115 a - 386\) , \( 1638 a - 5529\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(115a-386\right){x}+1638a-5529$ |
| 588.3-b1 |
588.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
588.3 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{2} \cdot 7^{4} \) |
$2.52778$ |
$(-a+3), (-2a+7), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.504756683$ |
$6.807860938$ |
3.589112196 |
\( \frac{46657}{147} a + \frac{31741}{49} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 3 a + 7\) , \( 5 a + 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(3a+7\right){x}+5a+11$ |
| 588.3-b2 |
588.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
588.3 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
$2.52778$ |
$(-a+3), (-2a+7), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1.009513367$ |
$13.61572187$ |
3.589112196 |
\( -\frac{1152541}{21} a + \frac{3923189}{21} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 96 a - 319\) , \( 838 a - 2824\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(96a-319\right){x}+838a-2824$ |
| 588.3-c1 |
588.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
588.3 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{2} \cdot 7^{4} \) |
$2.52778$ |
$(-a+3), (-2a+7), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.473663712$ |
$13.45052369$ |
2.218106190 |
\( \frac{46657}{147} a + \frac{31741}{49} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 138 a - 473\) , \( -6203 a + 20914\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(138a-473\right){x}-6203a+20914$ |
| 588.3-c2 |
588.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
588.3 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
$2.52778$ |
$(-a+3), (-2a+7), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.947327424$ |
$26.90104738$ |
2.218106190 |
\( -\frac{1152541}{21} a + \frac{3923189}{21} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -10 a - 20\) , \( -9 a - 20\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a-20\right){x}-9a-20$ |
| 588.3-d1 |
588.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
588.3 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{11} \cdot 7^{2} \) |
$2.52778$ |
$(-a+3), (-2a+7), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$5.010563700$ |
0.872227183 |
\( \frac{1904464}{5103} a - \frac{7326080}{5103} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( -23 a - 53\) , \( -369 a - 876\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-23a-53\right){x}-369a-876$ |
*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.
