| 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 |
| 79.2-a1 |
79.2-a |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( 79^{2} \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$111.2348527$ |
1.243643460 |
\( -\frac{379106265967104}{6241} a^{3} + \frac{721103039411200}{6241} a^{2} + \frac{523911972587520}{6241} a - \frac{996539889157440}{6241} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - 3\) , \( a^{2} + a - 2\) , \( 3 a^{3} - 6 a^{2} - 9 a + 15\) , \( -4 a^{3} + 2 a^{2} + 18 a - 17\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(3a^{3}-6a^{2}-9a+15\right){x}-4a^{3}+2a^{2}+18a-17$ |
| 79.2-a2 |
79.2-a |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( 79^{4} \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$111.2348527$ |
1.243643460 |
\( \frac{2979736840704}{38950081} a^{3} - \frac{6690282255360}{38950081} a^{2} - \frac{3428388587520}{38950081} a + \frac{10221062240960}{38950081} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + 3\) , \( a^{3} - 2 a\) , \( a^{2} - 3\) , \( -5 a^{3} - 9 a^{2} + 8 a + 11\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(a^{2}-3\right){x}-5a^{3}-9a^{2}+8a+11$ |
| 79.2-b1 |
79.2-b |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( 79^{2} \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$155.2093026$ |
1.735292756 |
\( -\frac{84090602204552041}{6241} a^{3} - \frac{98854431667475790}{6241} a^{2} + \frac{304242656911637730}{6241} a + \frac{357658693727064250}{6241} \) |
\( \bigl[a + 1\) , \( -a^{2} - a + 2\) , \( 0\) , \( -4 a^{3} + 12 a - 8\) , \( -3 a^{3} + 10 a^{2} + 16 a - 28\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-4a^{3}+12a-8\right){x}-3a^{3}+10a^{2}+16a-28$ |
| 79.2-b2 |
79.2-b |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( -79 \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$310.4186052$ |
1.735292756 |
\( -\frac{126473456}{79} a^{3} - \frac{148661440}{79} a^{2} + \frac{457577700}{79} a + \frac{537956405}{79} \) |
\( \bigl[a + 1\) , \( -a^{2} - a + 2\) , \( 0\) , \( a^{3} - 3 a + 2\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(a^{3}-3a+2\right){x}$ |
| 79.2-c1 |
79.2-c |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( 79^{2} \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.034185335$ |
$921.5828295$ |
1.408929375 |
\( -\frac{84090602204552041}{6241} a^{3} - \frac{98854431667475790}{6241} a^{2} + \frac{304242656911637730}{6241} a + \frac{357658693727064250}{6241} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 3\) , \( 1\) , \( a^{3} + a^{2} - 2 a - 3\) , \( -4 a^{3} + 12 a - 9\) , \( -a^{3} - 10 a^{2} - 4 a + 19\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-3\right){x}{y}+\left(a^{3}+a^{2}-2a-3\right){y}={x}^{3}+{x}^{2}+\left(-4a^{3}+12a-9\right){x}-a^{3}-10a^{2}-4a+19$ |
| 79.2-c2 |
79.2-c |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( -79 \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.068370671$ |
$921.5828295$ |
1.408929375 |
\( -\frac{126473456}{79} a^{3} - \frac{148661440}{79} a^{2} + \frac{457577700}{79} a + \frac{537956405}{79} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 3\) , \( 1\) , \( a^{3} + a^{2} - 2 a - 3\) , \( a^{3} - 3 a + 1\) , \( a^{3} - 3 a + 1\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-3\right){x}{y}+\left(a^{3}+a^{2}-2a-3\right){y}={x}^{3}+{x}^{2}+\left(a^{3}-3a+1\right){x}+a^{3}-3a+1$ |
| 79.2-d1 |
79.2-d |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( 79^{2} \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.069144720$ |
$582.6052374$ |
1.801558660 |
\( -\frac{379106265967104}{6241} a^{3} + \frac{721103039411200}{6241} a^{2} + \frac{523911972587520}{6241} a - \frac{996539889157440}{6241} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} - a^{2} + 3 a + 4\) , \( a^{3} - 2 a\) , \( 3 a^{3} - 9 a^{2} - 9 a + 25\) , \( 7 a^{3} - 10 a^{2} - 27 a + 37\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+4\right){x}^{2}+\left(3a^{3}-9a^{2}-9a+25\right){x}+7a^{3}-10a^{2}-27a+37$ |
| 79.2-d2 |
79.2-d |
$2$ |
$2$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
79.2 |
\( 79 \) |
\( 79^{4} \) |
$6.90012$ |
$(a^3+2a^2-4a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.034572360$ |
$582.6052374$ |
1.801558660 |
\( \frac{2979736840704}{38950081} a^{3} - \frac{6690282255360}{38950081} a^{2} - \frac{3428388587520}{38950081} a + \frac{10221062240960}{38950081} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( a^{3} + 4 a^{2} - 4 a - 11\) , \( 6 a^{3} + 11 a^{2} - 11 a - 18\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{3}+4a^{2}-4a-11\right){x}+6a^{3}+11a^{2}-11a-18$ |
*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.
