| 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 |
| 52.1-a1 |
52.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{8} \cdot 13^{2} \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$8.301308337$ |
1.198190650 |
\( -\frac{233472}{169} a - \frac{397312}{169} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 2\) , \( -2 a + 3\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+2{x}-2a+3$ |
| 52.1-a2 |
52.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{8} \cdot 13^{6} \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$8.301308337$ |
1.198190650 |
\( \frac{3013704404992}{4826809} a - \frac{5212493443072}{4826809} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 60 a - 98\) , \( -306 a + 527\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(60a-98\right){x}-306a+527$ |
| 52.1-a3 |
52.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{4} \cdot 13^{3} \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$16.60261667$ |
1.198190650 |
\( -\frac{2964729839810656}{2197} a + \frac{5135062717540112}{2197} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 3 a - 53\) , \( -61 a + 76\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-53\right){x}-61a+76$ |
| 52.1-a4 |
52.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{4} \cdot 13 \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$16.60261667$ |
1.198190650 |
\( \frac{111501408}{13} a + \frac{193353296}{13} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -7 a - 13\) , \( -29 a - 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-7a-13\right){x}-29a-50$ |
| 52.1-b1 |
52.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{8} \cdot 13^{2} \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$14.68942809$ |
0.706745438 |
\( -\frac{233472}{169} a - \frac{397312}{169} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 2\) , \( 2 a - 3\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+2{x}+2a-3$ |
| 52.1-b2 |
52.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{8} \cdot 13^{6} \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.632158677$ |
0.706745438 |
\( \frac{3013704404992}{4826809} a - \frac{5212493443072}{4826809} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 60 a - 98\) , \( 306 a - 527\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(60a-98\right){x}+306a-527$ |
| 52.1-b3 |
52.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{4} \cdot 13^{3} \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$3.264317354$ |
0.706745438 |
\( -\frac{2964729839810656}{2197} a + \frac{5135062717540112}{2197} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 2 a - 54\) , \( 9 a - 124\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(2a-54\right){x}+9a-124$ |
| 52.1-b4 |
52.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
52.1 |
\( 2^{2} \cdot 13 \) |
\( 2^{4} \cdot 13 \) |
$0.83125$ |
$(a+1), (a+4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$29.37885619$ |
0.706745438 |
\( \frac{111501408}{13} a + \frac{193353296}{13} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -8 a - 14\) , \( 7 a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-8a-14\right){x}+7a+12$ |
*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.
