| 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 |
| 15.1-a1 |
15.1-a |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$1.884794823$ |
0.940562166 |
\( \frac{4561952344347894508816}{50625} a^{2} + \frac{5468372973632837498513}{50625} a - \frac{6224545788789433223542}{50625} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -80 a^{2} - 340 a - 369\) , \( -2244 a^{2} - 5118 a - 1584\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-80a^{2}-340a-369\right){x}-2244a^{2}-5118a-1584$ |
| 15.1-a2 |
15.1-a |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{2} \cdot 5^{2} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$60.31343436$ |
0.940562166 |
\( \frac{25393788989964280607}{225} a^{2} - \frac{73952533248919150724}{225} a + \frac{39839035941583441516}{225} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 5 a^{2} + 20 a - 64\) , \( -2 a^{2} - 91 a + 193\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a^{2}+20a-64\right){x}-2a^{2}-91a+193$ |
| 15.1-a3 |
15.1-a |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$15.07835859$ |
0.940562166 |
\( \frac{311251219371829121}{2562890625} a^{2} + \frac{373240221797370628}{2562890625} a - \frac{424319657035623452}{2562890625} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -5 a^{2} - 20 a - 24\) , \( -44 a^{2} - 93 a - 9\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a^{2}-20a-24\right){x}-44a^{2}-93a-9$ |
| 15.1-a4 |
15.1-a |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$120.6268687$ |
0.940562166 |
\( \frac{5176756484309}{50625} a^{2} - \frac{15075007347263}{50625} a + \frac{8121003692917}{50625} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -4\) , \( -a^{2} - 4 a + 4\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}-4{x}-a^{2}-4a+4$ |
| 15.1-a5 |
15.1-a |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{16} \cdot 5^{16} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.884794823$ |
0.940562166 |
\( -\frac{28443849277171190300944}{6568408355712890625} a^{2} - \frac{34194402733320830270417}{6568408355712890625} a + \frac{38637799905139432891078}{6568408355712890625} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -10 a^{2} - 20 a + 1\) , \( -16 a^{2} - 104 a - 126\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-10a^{2}-20a+1\right){x}-16a^{2}-104a-126$ |
| 15.1-a6 |
15.1-a |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$241.2537374$ |
0.940562166 |
\( -\frac{660182}{225} a^{2} + \frac{2137499}{225} a - \frac{807616}{225} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+{x}$ |
| 15.1-b1 |
15.1-b |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{2} \cdot 5^{2} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$4.387233228$ |
1.094672359 |
\( \frac{25393788989964280607}{225} a^{2} - \frac{73952533248919150724}{225} a + \frac{39839035941583441516}{225} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 1\) , \( -66 a^{2} + 195 a - 107\) , \( -762 a^{2} + 2197 a - 1180\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-66a^{2}+195a-107\right){x}-762a^{2}+2197a-1180$ |
| 15.1-b2 |
15.1-b |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$35.09786582$ |
1.094672359 |
\( \frac{5176756484309}{50625} a^{2} - \frac{15075007347263}{50625} a + \frac{8121003692917}{50625} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 1\) , \( -6 a^{2} + 10 a - 2\) , \( -16 a^{2} + 41 a - 22\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-6a^{2}+10a-2\right){x}-16a^{2}+41a-22$ |
| 15.1-b3 |
15.1-b |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$70.19573165$ |
1.094672359 |
\( -\frac{660182}{225} a^{2} + \frac{2137499}{225} a - \frac{807616}{225} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 1\) , \( -a^{2} + 3\) , \( a - 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-a^{2}+3\right){x}+a-2$ |
| 15.1-b4 |
15.1-b |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$17.54893291$ |
1.094672359 |
\( \frac{311251219371829121}{2562890625} a^{2} + \frac{373240221797370628}{2562890625} a - \frac{424319657035623452}{2562890625} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 1\) , \( -26 a^{2} - 15 a + 23\) , \( -114 a^{2} - 75 a + 116\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-26a^{2}-15a+23\right){x}-114a^{2}-75a+116$ |
| 15.1-b5 |
15.1-b |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{16} \cdot 5^{16} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$2.193616614$ |
1.094672359 |
\( -\frac{28443849277171190300944}{6568408355712890625} a^{2} - \frac{34194402733320830270417}{6568408355712890625} a + \frac{38637799905139432891078}{6568408355712890625} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 1\) , \( -21 a^{2} - 30 a + 33\) , \( -136 a^{2} - 19 a + 88\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-21a^{2}-30a+33\right){x}-136a^{2}-19a+88$ |
| 15.1-b6 |
15.1-b |
$6$ |
$8$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$2.24968$ |
$(a^2-3), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$8.774466456$ |
1.094672359 |
\( \frac{4561952344347894508816}{50625} a^{2} + \frac{5468372973632837498513}{50625} a - \frac{6224545788789433223542}{50625} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 1\) , \( -351 a^{2} - 400 a + 413\) , \( -6364 a^{2} - 7615 a + 8876\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-351a^{2}-400a+413\right){x}-6364a^{2}-7615a+8876$ |
*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.
