| 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 |
| 22.1-b3 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{10} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2Cs, 5B.1.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$1697.894796$ |
1.610989990 |
\( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 4 a^{3} - 7 a^{2} - 9 a + 5\) , \( -a^{3} + 2 a^{2} + a - 3\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(4a^{3}-7a^{2}-9a+5\right){x}-a^{3}+2a^{2}+a-3$ |
| 176.2-c6 |
176.2-c |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
176.2 |
\( 2^{4} \cdot 11 \) |
\( 2^{22} \cdot 11^{2} \) |
$8.98702$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2Cs, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$165.5951851$ |
1.571193846 |
\( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) |
\( \bigl[a\) , \( -a^{2} + 3\) , \( a^{3} - 3 a\) , \( -a^{3} - a^{2} + 4 a - 1\) , \( -3 a^{3} + 3 a^{2} + 9 a - 9\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{3}-a^{2}+4a-1\right){x}-3a^{3}+3a^{2}+9a-9$ |
| 242.1-a4 |
242.1-a |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
242.1 |
\( 2 \cdot 11^{2} \) |
\( 2^{10} \cdot 11^{8} \) |
$9.35198$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2Cs, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$0.397322195$ |
$141.1320749$ |
2.128191031 |
\( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 4 a - 2\) , \( a^{3} - a^{2} - 2 a + 1\) , \( 5 a^{3} + 8 a^{2} - 32 a - 45\) , \( 17 a^{3} - 18 a^{2} - 63 a + 4\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(5a^{3}+8a^{2}-32a-45\right){x}+17a^{3}-18a^{2}-63a+4$ |
| 704.3-g5 |
704.3-g |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{28} \cdot 11^{2} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2Cs, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$0.355724909$ |
$258.6792253$ |
3.492350992 |
\( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - a - 2\) , \( a^{2} - 2\) , \( a^{3} - 7 a^{2} + 4\) , \( 2 a^{3} - 3 a^{2} + 2 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{3}-7a^{2}+4\right){x}+2a^{3}-3a^{2}+2a-1$ |
| 704.3-n7 |
704.3-n |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
704.3 |
\( 2^{6} \cdot 11 \) |
\( 2^{28} \cdot 11^{2} \) |
$10.68742$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2Cs, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$178.4730992$ |
1.693381574 |
\( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) |
\( \bigl[a\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -27 a^{3} - 40 a^{2} + 15 a + 22\) , \( -238 a^{3} - 327 a^{2} + 177 a + 194\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-27a^{3}-40a^{2}+15a+22\right){x}-238a^{3}-327a^{2}+177a+194$ |
*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.
