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 |
2.2-b1 |
2.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$18.51817441$ |
2.561857817 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 166348 a - 1285588\) , \( 101485499 a - 784322149\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(166348a-1285588\right){x}+101485499a-784322149$ |
2.2-c1 |
2.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.529462610$ |
$13.03498084$ |
1.909556628 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -1476 a - 9940\) , \( -95638 a - 643494\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1476a-9940\right){x}-95638a-643494$ |
50.3-a1 |
50.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a+31)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \) |
$0.129561378$ |
$8.281579361$ |
2.375010653 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}-a{x}$ |
50.3-g1 |
50.3-g |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a+31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \) |
$1.786022418$ |
$5.829420649$ |
11.52282962 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 557954489 a - 4312104315\) , \( -19506746796820 a + 150756256928227\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(557954489a-4312104315\right){x}-19506746796820a+150756256928227$ |
50.4-b1 |
50.4-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.4 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a-27)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$7.111946551$ |
$8.281579361$ |
16.29628084 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -4988005 a - 33561363\) , \( 16267292978 a + 109453116751\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-4988005a-33561363\right){x}+16267292978a+109453116751$ |
50.4-k1 |
50.4-k |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.4 |
\( 2 \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a-27)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$5.829420649$ |
0.806458915 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 1241 a - 9560\) , \( -60361 a + 466576\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(1241a-9560\right){x}-60361a+466576$ |
64.6-c1 |
64.6-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{20} \) |
$3.65390$ |
$(11a-85)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
|
\( 2 \) |
$1$ |
$1.288073535$ |
7.117173427 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( -50 a - 326\) , \( -388 a - 2613\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-50a-326\right){x}-388a-2613$ |
64.6-g1 |
64.6-g |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{20} \) |
$3.65390$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.271303162$ |
$23.42490957$ |
1.758407907 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 310246949 a - 2397717530\) , \( -8088584565218 a + 62511947563426\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(310246949a-2397717530\right){x}-8088584565218a+62511947563426$ |
*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.