| 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 |
| 91.1-a1 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( 7^{16} \cdot 13^{2} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.284767587$ |
0.650897342 |
\( \frac{3586352161337298910516}{19882681} a^{2} - \frac{8058460190096647498093}{19882681} a + \frac{2876031146228402085760}{19882681} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -1030 a^{2} + 1620 a - 484\) , \( -21769 a^{2} + 41147 a - 14213\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-1030a^{2}+1620a-484\right){x}-21769a^{2}+41147a-14213$ |
| 91.1-a2 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( 7 \cdot 13^{2} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$145.8010047$ |
0.650897342 |
\( -\frac{929922096412289245}{1183} a^{2} + \frac{516067794318659937}{1183} a + \frac{2089516047340720303}{1183} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( 15 a^{2} + 5 a - 89\) , \( -89 a^{2} - 20 a + 385\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(15a^{2}+5a-89\right){x}-89a^{2}-20a+385$ |
| 91.1-a3 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( 7^{2} \cdot 13^{4} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$145.8010047$ |
0.650897342 |
\( -\frac{3698907677516}{199927} a^{2} + \frac{293174005427}{28561} a + \frac{8312780816110}{199927} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -4\) , \( -3 a^{2} - a + 9\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}-4{x}-3a^{2}-a+9$ |
| 91.1-a4 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( 7^{4} \cdot 13^{8} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$18.22512558$ |
0.650897342 |
\( \frac{20917603896641523}{39970805329} a^{2} + \frac{16797981605493841}{39970805329} a - \frac{11327303846528113}{39970805329} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -15 a^{2} - 5 a + 1\) , \( -49 a^{2} - 26 a + 29\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-15a^{2}-5a+1\right){x}-49a^{2}-26a+29$ |
| 91.1-a5 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( - 7 \cdot 13^{2} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$145.8010047$ |
0.650897342 |
\( \frac{3379823}{1183} a^{2} - \frac{1448892}{1183} a - \frac{6890188}{1183} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+{x}$ |
| 91.1-a6 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( - 7^{2} \cdot 13^{16} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$2.278140698$ |
0.650897342 |
\( -\frac{1202964810122504163047}{4657916264282258887} a^{2} - \frac{166312704815995127056}{665416609183179841} a + \frac{2026661189545263268136}{4657916264282258887} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -10 a^{2} - 20 a + 21\) , \( -90 a^{2} + 20 a + 26\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-10a^{2}-20a+21\right){x}-90a^{2}+20a+26$ |
| 91.1-a7 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( 7^{8} \cdot 13^{4} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$2.278140698$ |
0.650897342 |
\( \frac{11174818063860788327}{9796423} a^{2} + \frac{8961331728253148016}{9796423} a - \frac{6201477492864641144}{9796423} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -260 a^{2} - 70 a + 61\) , \( -2872 a^{2} - 1452 a + 1192\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-260a^{2}-70a+61\right){x}-2872a^{2}-1452a+1192$ |
| 91.1-a8 |
91.1-a |
$8$ |
$16$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
91.1 |
\( 7 \cdot 13 \) |
\( - 7^{4} \cdot 13^{2} \) |
$1.32661$ |
$(-a^2-a+2), (-2a^2+a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.284767587$ |
0.650897342 |
\( \frac{44577562611830213731988085324}{8281} a^{2} + \frac{35748429628629533712071020429}{8281} a - \frac{24738680880069130876826540592}{8281} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 3\) , \( 0\) , \( -3410 a^{2} - 2800 a + 1566\) , \( -163347 a^{2} - 131435 a + 88489\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-3410a^{2}-2800a+1566\right){x}-163347a^{2}-131435a+88489$ |
*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.
