| 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 |
| 2.1-b1 |
2.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a+74)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$18.51817441$ |
2.561857817 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -166349 a - 1119240\) , \( -101485499 a - 682836650\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-166349a-1119240\right){x}-101485499a-682836650$ |
| 2.1-c1 |
2.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.529462610$ |
$13.03498084$ |
1.909556628 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 1502 a - 11391\) , \( 84221 a - 650261\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1502a-11391\right){x}+84221a-650261$ |
| 50.5-b1 |
50.5-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.5 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a+74), (-4a+31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$7.111946551$ |
$8.281579361$ |
16.29628084 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 4988033 a - 38549342\) , \( -16300854340 a + 125979786691\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4988033a-38549342\right){x}-16300854340a+125979786691$ |
| 50.5-k1 |
50.5-k |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.5 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a+74), (-4a+31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$5.829420649$ |
0.806458915 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -1243 a - 8318\) , \( 60360 a + 406216\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1243a-8318\right){x}+60360a+406216$ |
| 50.6-a1 |
50.6-a |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.6 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a+74), (-4a-27)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \) |
$0.129561378$ |
$8.281579361$ |
2.375010653 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 27 a + 25\) , \( 702\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(27a+25\right){x}+702$ |
| 50.6-g1 |
50.6-g |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.6 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a+74), (-4a-27)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \) |
$1.786022418$ |
$5.829420649$ |
11.52282962 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -557954461 a - 3754149800\) , \( 19502434692506 a + 131220496498681\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-557954461a-3754149800\right){x}+19502434692506a+131220496498681$ |
| 64.7-c1 |
64.7-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{20} \) |
$3.65390$ |
$(11a+74)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
|
\( 2 \) |
$1$ |
$1.288073535$ |
7.117173427 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( 52 a - 377\) , \( 438 a - 3377\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(52a-377\right){x}+438a-3377$ |
| 64.7-g1 |
64.7-g |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{20} \) |
$3.65390$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.271303162$ |
$23.42490957$ |
1.758407907 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -310246949 a - 2087470581\) , \( 8088584565217 a + 54423362998209\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-310246949a-2087470581\right){x}+8088584565217a+54423362998209$ |
*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.
