| 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 |
| 1.1-a1 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1018.917972$ |
0.225151189 |
\( 5699182696 a^{3} - 4361830336 a^{2} - 19458081104 a + 14892491272 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} - a + 3\) , \( 1\) , \( a^{3} - 4 a\) , \( -22 a^{3} - 17 a^{2} + 75 a + 58\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(a^{3}-4a\right){x}-22a^{3}-17a^{2}+75a+58$ |
| 1.1-a2 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2Cs, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$6.521075021$ |
0.225151189 |
\( 75602392581248 a^{2} - 44286841602368 \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - 1\) , \( 120 a^{3} - 250 a^{2} + 40 a + 19\) , \( 2105 a^{3} - 4179 a^{2} - 366 a + 1660\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(120a^{3}-250a^{2}+40a+19\right){x}+2105a^{3}-4179a^{2}-366a+1660$ |
| 1.1-a3 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2Cs, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$4075.671888$ |
0.225151189 |
\( -55168 a^{2} + 190144 \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -2 a^{3} - 2 a^{2} + 4 a + 3\) , \( -a^{3} - 2 a^{2} + a + 2\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(-2a^{3}-2a^{2}+4a+3\right){x}-a^{3}-2a^{2}+a+2$ |
| 1.1-a4 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2Cs, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$6.521075021$ |
0.225151189 |
\( -75602392581248 a^{2} + 258122728722624 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 2 a + 3\) , \( a + 1\) , \( 120 a^{3} + 108 a^{2} - 523 a - 576\) , \( 1768 a^{3} + 1361 a^{2} - 7025 a - 6474\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+3\right){x}^{2}+\left(120a^{3}+108a^{2}-523a-576\right){x}+1768a^{3}+1361a^{2}-7025a-6474$ |
| 1.1-a5 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2Cs, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$4075.671888$ |
0.225151189 |
\( 55168 a^{2} - 30528 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 2 a + 3\) , \( a + 1\) , \( -2 a^{2} - 3 a + 4\) , \( a^{3} - 3 a\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+3\right){x}^{2}+\left(-2a^{2}-3a+4\right){x}+a^{3}-3a$ |
| 1.1-a6 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1018.917972$ |
0.225151189 |
\( 2360533016 a^{3} + 4361830336 a^{2} - 1382416352 a - 2554830072 \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} - a - 1\) , \( 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -9 a^{3} + 17 a^{2} + 5 a - 10\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(a^{3}-a^{2}-2a+2\right){x}-9a^{3}+17a^{2}+5a-10$ |
| 1.1-a7 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1018.917972$ |
0.225151189 |
\( -2360533016 a^{3} + 4361830336 a^{2} + 1382416352 a - 2554830072 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 1\) , \( -a^{3} - a^{2} + a + 2\) , \( 9 a^{3} + 17 a^{2} - 5 a - 10\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-a^{3}-a^{2}+a+2\right){x}+9a^{3}+17a^{2}-5a-10$ |
| 1.1-a8 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1018.917972$ |
0.225151189 |
\( -5699182696 a^{3} - 4361830336 a^{2} + 19458081104 a + 14892491272 \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} - 3 a\) , \( a^{3} - 2 a + 1\) , \( 4 a^{3} - 5 a + 3\) , \( 3 a^{3} - 2 a\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(4a^{3}-5a+3\right){x}+3a^{3}-2a$ |
| 1.1-a9 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$1.630268755$ |
0.225151189 |
\( 10561278147056904530419721864 a^{3} - 8083252342956303105729121856 a^{2} - 36058459085676274419182662496 a + 27597949777405506664334735112 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 2 a - 1\) , \( a + 1\) , \( 1096 a^{3} + 877 a^{2} - 3737 a - 3004\) , \( 30377 a^{3} + 23405 a^{2} - 103704 a - 79930\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a-1\right){x}^{2}+\left(1096a^{3}+877a^{2}-3737a-3004\right){x}+30377a^{3}+23405a^{2}-103704a-79930$ |
| 1.1-a10 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$1.630268755$ |
0.225151189 |
\( 4374624644505560827923496904 a^{3} + 8083252342956303105729121856 a^{2} - 2562595786459777953350768848 a - 4735059594419705758581752312 \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 1\) , \( -113 a^{3} + 112 a^{2} + 423 a - 455\) , \( -1225 a^{3} + 1100 a^{2} + 4334 a - 4038\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-113a^{3}+112a^{2}+423a-455\right){x}-1225a^{3}+1100a^{2}+4334a-4038$ |
| 1.1-a11 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$1.630268755$ |
0.225151189 |
\( -4374624644505560827923496904 a^{3} + 8083252342956303105729121856 a^{2} + 2562595786459777953350768848 a - 4735059594419705758581752312 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( 1\) , \( 112 a^{3} + 112 a^{2} - 420 a - 455\) , \( 1225 a^{3} + 1100 a^{2} - 4334 a - 4038\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(112a^{3}+112a^{2}-420a-455\right){x}+1225a^{3}+1100a^{2}-4334a-4038$ |
| 1.1-a12 |
1.1-a |
$12$ |
$40$ |
\(\Q(\zeta_{16})^+\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.04393$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$1.630268755$ |
0.225151189 |
\( -10561278147056904530419721864 a^{3} - 8083252342956303105729121856 a^{2} + 36058459085676274419182662496 a + 27597949777405506664334735112 \) |
\( \bigl[a\) , \( a^{2} + a - 2\) , \( 1\) , \( -84 a^{3} - 112 a^{2} + 139 a - 7\) , \( -659 a^{3} - 1100 a^{2} + 752 a + 362\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(-84a^{3}-112a^{2}+139a-7\right){x}-659a^{3}-1100a^{2}+752a+362$ |
*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.
