| 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 |
| 50.1-a1 |
50.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{8} \) |
$1.50283$ |
$(2,a), (5,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.2, 5B.1.3 |
$1$ |
\( 2 \) |
$6.862779968$ |
$0.508604290$ |
2.207547668 |
\( -\frac{349938025}{8} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -126\) , \( -552\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-126{x}-552$ |
| 50.1-a2 |
50.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{22} \cdot 5^{4} \) |
$1.50283$ |
$(2,a), (5,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.1, 5B.1.4 |
$1$ |
\( 2 \cdot 3 \) |
$0.457518664$ |
$22.88719308$ |
2.207547668 |
\( -\frac{121945}{32} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -10\) , \( 10\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-10{x}+10$ |
| 50.1-a3 |
50.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{8} \) |
$1.50283$ |
$(2,a), (5,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.1, 5B.1.3 |
$1$ |
\( 2 \cdot 3 \) |
$2.287593322$ |
$4.577438616$ |
2.207547668 |
\( -\frac{25}{2} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( -2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}-2$ |
| 50.1-a4 |
50.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{42} \cdot 5^{4} \) |
$1.50283$ |
$(2,a), (5,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B.1.2, 5B.1.4 |
$1$ |
\( 2 \) |
$1.372555993$ |
$2.543021453$ |
2.207547668 |
\( \frac{46969655}{32768} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 90\) , \( -70\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+90{x}-70$ |
| 50.1-b1 |
50.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{18} \cdot 5^{8} \) |
$1.50283$ |
$(2,a), (5,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.508604290$ |
1.447513187 |
\( -\frac{349938025}{8} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -503\) , \( -4917\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-503{x}-4917$ |
| 50.1-b2 |
50.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{4} \) |
$1.50283$ |
$(2,a), (5,a)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$22.88719308$ |
1.447513187 |
\( -\frac{121945}{32} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-3{x}+1$ |
| 50.1-b3 |
50.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{14} \cdot 5^{8} \) |
$1.50283$ |
$(2,a), (5,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$4.577438616$ |
1.447513187 |
\( -\frac{25}{2} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -3\) , \( -17\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-3{x}-17$ |
| 50.1-b4 |
50.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{30} \cdot 5^{4} \) |
$1.50283$ |
$(2,a), (5,a)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$2.543021453$ |
1.447513187 |
\( \frac{46969655}{32768} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 22\) , \( -9\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+22{x}-9$ |
*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.
