| 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 |
| 75.1-a1 |
75.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{8} \) |
$2.03695$ |
$(3,a), (5,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$21.80453609$ |
5.629907010 |
\( -\frac{102400}{3} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -8\) , \( 3\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}-8{x}+3$ |
| 75.1-a2 |
75.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 2^{12} \cdot 3^{10} \cdot 5^{4} \) |
$2.03695$ |
$(3,a), (5,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.360907218$ |
5.629907010 |
\( \frac{20480}{243} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -54 a + 212\) , \( 1756 a - 6802\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-54a+212\right){x}+1756a-6802$ |
| 75.1-b1 |
75.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{8} \) |
$2.03695$ |
$(3,a), (5,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 3 \) |
$0.025588090$ |
$21.80453609$ |
0.864351418 |
\( -\frac{102400}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -266 a - 1028\) , \( 5216 a + 20202\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-266a-1028\right){x}+5216a+20202$ |
| 75.1-b2 |
75.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{10} \cdot 5^{4} \) |
$2.03695$ |
$(3,a), (5,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.127940452$ |
$4.360907218$ |
0.864351418 |
\( \frac{20480}{243} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 2\) , \( -8\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+2{x}-8$ |
| 75.1-c1 |
75.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{8} \) |
$2.03695$ |
$(3,a), (5,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.967118283$ |
1.523723270 |
\( -\frac{102400}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -266 a - 1028\) , \( -5216 a - 20202\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-266a-1028\right){x}-5216a-20202$ |
| 75.1-c2 |
75.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{10} \cdot 5^{4} \) |
$2.03695$ |
$(3,a), (5,a)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$1$ |
$9.835591419$ |
1.523723270 |
\( \frac{20480}{243} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 2\) , \( 4\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+2{x}+4$ |
| 75.1-d1 |
75.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{8} \) |
$2.03695$ |
$(3,a), (5,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.1.3 |
$1$ |
\( 2 \) |
$4.591654055$ |
$1.967118283$ |
4.664273423 |
\( -\frac{102400}{3} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -8\) , \( -7\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-8{x}-7$ |
| 75.1-d2 |
75.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 2^{12} \cdot 3^{10} \cdot 5^{4} \) |
$2.03695$ |
$(3,a), (5,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.1.4 |
$1$ |
\( 2 \) |
$0.918330811$ |
$9.835591419$ |
4.664273423 |
\( \frac{20480}{243} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -54 a + 212\) , \( -1756 a + 6802\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-54a+212\right){x}-1756a+6802$ |
*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.
