| 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 |
| 45.2-a1 |
45.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{4} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$173.0488798$ |
1.199388060 |
\( \frac{266828773549}{625} a^{2} + \frac{319939354557}{625} a - \frac{363891969838}{625} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 4 a - 15\) , \( -2 a^{2} - 4 a + 19\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(4a-15\right){x}-2a^{2}-4a+19$ |
| 45.2-a2 |
45.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{2} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$173.0488798$ |
1.199388060 |
\( \frac{214892}{25} a^{2} + \frac{248781}{25} a - \frac{290629}{25} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -a\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}-a+1$ |
| 45.2-a3 |
45.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{12} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$19.22765331$ |
1.199388060 |
\( \frac{72134063979419}{244140625} a^{2} - \frac{211425954456408}{244140625} a + \frac{116702222801597}{244140625} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 4\) , \( a^{2} + a - 3\) , \( -312 a^{2} + 910 a - 489\) , \( -6763 a^{2} + 19697 a - 10612\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-312a^{2}+910a-489\right){x}-6763a^{2}+19697a-10612$ |
| 45.2-a4 |
45.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{6} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$19.22765331$ |
1.199388060 |
\( \frac{141845203}{15625} a^{2} + \frac{177714129}{15625} a - \frac{196843186}{15625} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 4\) , \( a^{2} + a - 3\) , \( -17 a^{2} + 50 a - 24\) , \( -127 a^{2} + 372 a - 202\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-17a^{2}+50a-24\right){x}-127a^{2}+372a-202$ |
| 45.2-b1 |
45.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{4} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.065907809$ |
$67.02863078$ |
1.653415156 |
\( \frac{266828773549}{625} a^{2} + \frac{319939354557}{625} a - \frac{363891969838}{625} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 3\) , \( 36 a^{2} - 12 a - 156\) , \( 166 a^{2} - 48 a - 700\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}-{x}^{2}+\left(36a^{2}-12a-156\right){x}+166a^{2}-48a-700$ |
| 45.2-b2 |
45.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{2} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.131815618$ |
$67.02863078$ |
1.653415156 |
\( \frac{214892}{25} a^{2} + \frac{248781}{25} a - \frac{290629}{25} \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} + a - 3\) , \( a^{2} - 2 a - 6\) , \( 4 a^{2} - 2 a - 19\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}-{x}^{2}+\left(a^{2}-2a-6\right){x}+4a^{2}-2a-19$ |
| 45.2-b3 |
45.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{12} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.021969269$ |
$67.02863078$ |
1.653415156 |
\( \frac{72134063979419}{244140625} a^{2} - \frac{211425954456408}{244140625} a + \frac{116702222801597}{244140625} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} - 4\) , \( a + 1\) , \( -32 a^{2} + 108 a - 74\) , \( 277 a^{2} - 829 a + 494\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-32a^{2}+108a-74\right){x}+277a^{2}-829a+494$ |
| 45.2-b4 |
45.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{6} \) |
$2.70172$ |
$(a^2-3), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.043938539$ |
$67.02863078$ |
1.653415156 |
\( \frac{141845203}{15625} a^{2} + \frac{177714129}{15625} a - \frac{196843186}{15625} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} - 4\) , \( a + 1\) , \( -2 a^{2} + 8 a + 1\) , \( 8 a^{2} - 15 a + 5\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-2a^{2}+8a+1\right){x}+8a^{2}-15a+5$ |
*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.
