| 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 |
| 3.1-a1 |
3.1-a |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{8} \) |
$10.20454$ |
$(-a^3+a^2+4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$140.0282857$ |
0.703348980 |
\( -357531767 a^{3} + \frac{3885538150}{9} a^{2} + \frac{14614849117}{9} a - \frac{7994350247}{9} \) |
\( \bigl[a^{3} - 4 a - 2\) , \( a^{2} - 2 a - 2\) , \( a^{3} - 5 a - 1\) , \( 13 a^{3} - 28 a^{2} - 26 a + 32\) , \( -85 a^{3} + 160 a^{2} + 209 a - 133\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(13a^{3}-28a^{2}-26a+32\right){x}-85a^{3}+160a^{2}+209a-133$ |
| 3.1-a2 |
3.1-a |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{4} \) |
$10.20454$ |
$(-a^3+a^2+4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$140.0282857$ |
0.703348980 |
\( 568516610958557191 a^{3} - 686495379714342911 a^{2} - \frac{7746427564028264581}{3} a + \frac{4237319461104453760}{3} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 3\) , \( 358 a^{3} + 180 a^{2} - 2067 a - 2129\) , \( 836 a^{3} + 427 a^{2} - 4792 a - 4938\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(358a^{3}+180a^{2}-2067a-2129\right){x}+836a^{3}+427a^{2}-4792a-4938$ |
| 3.1-b1 |
3.1-b |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{4} \) |
$10.20454$ |
$(-a^3+a^2+4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$269.1804840$ |
1.352068391 |
\( 568516610958557191 a^{3} - 686495379714342911 a^{2} - \frac{7746427564028264581}{3} a + \frac{4237319461104453760}{3} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( 0\) , \( 60 a^{3} + 24 a^{2} - 336 a - 336\) , \( -6 a^{3} - 33 a^{2} + 75 a + 129\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(60a^{3}+24a^{2}-336a-336\right){x}-6a^{3}-33a^{2}+75a+129$ |
| 3.1-b2 |
3.1-b |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{8} \) |
$10.20454$ |
$(-a^3+a^2+4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$269.1804840$ |
1.352068391 |
\( -357531767 a^{3} + \frac{3885538150}{9} a^{2} + \frac{14614849117}{9} a - \frac{7994350247}{9} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( 0\) , \( -15 a^{3} - 6 a^{2} + 84 a + 84\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-15a^{3}-6a^{2}+84a+84\right){x}$ |
| 13.1-a1 |
13.1-a |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.079208123$ |
$441.3551652$ |
2.126649835 |
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -17 a^{3} - 3 a^{2} + 34 a - 30\) , \( 45 a^{3} + 106 a^{2} + 25 a - 23\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-17a^{3}-3a^{2}+34a-30\right){x}+45a^{3}+106a^{2}+25a-23$ |
| 13.1-a2 |
13.1-a |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13^{3} \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.359736041$ |
$441.3551652$ |
2.126649835 |
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -2 a^{3} + 2 a^{2} + 9 a - 5\) , \( -14 a^{3} + 17 a^{2} + 64 a - 35\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-2a^{3}+2a^{2}+9a-5\right){x}-14a^{3}+17a^{2}+64a-35$ |
| 13.1-a3 |
13.1-a |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1.079208123$ |
$5.448829201$ |
2.126649835 |
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a\) , \( 0\) , \( 21 a^{3} + 8 a^{2} - 26 a - 9\) , \( 562 a^{3} + 1305 a^{2} + 277 a - 581\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+a{x}^{2}+\left(21a^{3}+8a^{2}-26a-9\right){x}+562a^{3}+1305a^{2}+277a-581$ |
| 13.1-b1 |
13.1-b |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13^{3} \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.277561838$ |
$403.6980252$ |
1.500860066 |
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \) |
\( \bigl[a^{3} - 4 a - 2\) , \( a\) , \( a^{3} - 5 a - 1\) , \( 9 a^{3} - 18 a^{2} - 22 a + 14\) , \( -46 a^{3} + 74 a^{2} + 104 a - 66\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+a{x}^{2}+\left(9a^{3}-18a^{2}-22a+14\right){x}-46a^{3}+74a^{2}+104a-66$ |
| 13.1-b2 |
13.1-b |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$0.832685514$ |
$4.983926238$ |
1.500860066 |
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - a - 2\) , \( -130 a^{3} + 234 a^{2} + 353 a - 228\) , \( 477 a^{3} - 959 a^{2} - 995 a + 687\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-130a^{3}+234a^{2}+353a-228\right){x}+477a^{3}-959a^{2}-995a+687$ |
| 13.1-b3 |
13.1-b |
$3$ |
$9$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$12.25734$ |
$(-a^2+2a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.832685514$ |
$403.6980252$ |
1.500860066 |
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a\) , \( 1\) , \( -220 a^{3} + 274 a^{2} + 1002 a - 570\) , \( 2785 a^{3} - 3453 a^{2} - 12765 a + 7022\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-220a^{3}+274a^{2}+1002a-570\right){x}+2785a^{3}-3453a^{2}-12765a+7022$ |
| 15.1-a1 |
15.1-a |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$489.3919812$ |
1.229085070 |
\( -\frac{106752257}{9375} a^{3} - \frac{10897274}{3125} a^{2} + \frac{42742157}{625} a + \frac{211007117}{3125} \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - 2\) , \( 2 a^{3} + a^{2} - 11 a - 9\) , \( 3 a^{3} + a^{2} - 17 a - 16\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(2a^{3}+a^{2}-11a-9\right){x}+3a^{3}+a^{2}-17a-16$ |
| 15.1-a2 |
15.1-a |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{20} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$122.3479953$ |
1.229085070 |
\( -\frac{51150296581786130426003}{286102294921875} a^{3} + \frac{32230691277422593955829}{95367431640625} a^{2} + \frac{24823185420593409932434}{57220458984375} a - \frac{81174148800215168575921}{286102294921875} \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - 2\) , \( -3 a^{3} - 29 a^{2} + 39 a + 86\) , \( -53 a^{3} + 71 a^{2} + 190 a + 33\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-3a^{3}-29a^{2}+39a+86\right){x}-53a^{3}+71a^{2}+190a+33$ |
| 15.1-a3 |
15.1-a |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$122.3479953$ |
1.229085070 |
\( \frac{68545439115264271007}{9375} a^{3} + \frac{59190226096332625399}{3125} a^{2} + \frac{3249096321761237218}{625} a - \frac{26459436802864644617}{3125} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 3\) , \( -426 a^{3} + 526 a^{2} + 1926 a - 1114\) , \( -7407 a^{3} + 8927 a^{2} + 33648 a - 18305\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-426a^{3}+526a^{2}+1926a-1114\right){x}-7407a^{3}+8927a^{2}+33648a-18305$ |
| 15.1-a4 |
15.1-a |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{10} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$489.3919812$ |
1.229085070 |
\( \frac{422494591886387}{29296875} a^{3} + \frac{1087832991903677}{29296875} a^{2} + \frac{19383695526463}{1953125} a - \frac{160867452574272}{9765625} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 3\) , \( -26 a^{3} + 31 a^{2} + 121 a - 64\) , \( -120 a^{3} + 147 a^{2} + 543 a - 302\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-26a^{3}+31a^{2}+121a-64\right){x}-120a^{3}+147a^{2}+543a-302$ |
| 15.1-b1 |
15.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 5 \) |
$0.601291461$ |
$21.04466902$ |
2.542390272 |
\( \frac{68545439115264271007}{9375} a^{3} + \frac{59190226096332625399}{3125} a^{2} + \frac{3249096321761237218}{625} a - \frac{26459436802864644617}{3125} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -9 a^{3} + 22 a^{2} + a - 50\) , \( 1301 a^{3} - 2423 a^{2} - 3266 a + 1886\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-9a^{3}+22a^{2}+a-50\right){x}+1301a^{3}-2423a^{2}-3266a+1886$ |
| 15.1-b2 |
15.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.150322865$ |
$1346.858817$ |
2.542390272 |
\( -\frac{106752257}{9375} a^{3} - \frac{10897274}{3125} a^{2} + \frac{42742157}{625} a + \frac{211007117}{3125} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 6 a^{3} - 13 a^{2} - 14 a + 15\) , \( -26 a^{3} + 48 a^{2} + 64 a - 40\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(6a^{3}-13a^{2}-14a+15\right){x}-26a^{3}+48a^{2}+64a-40$ |
| 15.1-b3 |
15.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{10} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.300645730$ |
$168.3573522$ |
2.542390272 |
\( \frac{422494591886387}{29296875} a^{3} + \frac{1087832991903677}{29296875} a^{2} + \frac{19383695526463}{1953125} a - \frac{160867452574272}{9765625} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 51 a^{3} - 98 a^{2} - 124 a + 85\) , \( 296 a^{3} - 560 a^{2} - 720 a + 467\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(51a^{3}-98a^{2}-124a+85\right){x}+296a^{3}-560a^{2}-720a+467$ |
| 15.1-b4 |
15.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{20} \) |
$12.47857$ |
$(-a^3+a^2+4a), (-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.601291461$ |
$10.52233451$ |
2.542390272 |
\( -\frac{51150296581786130426003}{286102294921875} a^{3} + \frac{32230691277422593955829}{95367431640625} a^{2} + \frac{24823185420593409932434}{57220458984375} a - \frac{81174148800215168575921}{286102294921875} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 4\) , \( a^{2} - 2\) , \( -37 a^{3} + 19 a^{2} + 184 a + 26\) , \( -27 a^{3} + 43 a^{2} + 113 a - 124\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+4\right){x}^{2}+\left(-37a^{3}+19a^{2}+184a+26\right){x}-27a^{3}+43a^{2}+113a-124$ |
| 16.1-a1 |
16.1-a |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$12.57965$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.390079341$ |
$529.1940998$ |
2.073733912 |
\( -720868474875 a^{3} + \frac{2726844170253}{2} a^{2} + \frac{3499385683905}{2} a - 1144450579761 \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - a - 3\) , \( a^{2} - 3\) , \( 104 a^{3} + 58 a^{2} - 602 a - 641\) , \( -354 a^{3} - 185 a^{2} + 2035 a + 2117\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(104a^{3}+58a^{2}-602a-641\right){x}-354a^{3}-185a^{2}+2035a+2117$ |
| 16.1-a2 |
16.1-a |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.57965$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.195039670$ |
$529.1940998$ |
2.073733912 |
\( \frac{622203}{2} a^{3} - \frac{2391861}{4} a^{2} - \frac{3043377}{4} a + \frac{2003589}{4} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 2 a^{2} + 2 a - 4\) , \( a^{2} - a - 2\) , \( 2 a^{3} - 4 a^{2} - 6 a + 3\) , \( -a - 1\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-4\right){x}^{2}+\left(2a^{3}-4a^{2}-6a+3\right){x}-a-1$ |
| 16.1-b1 |
16.1-b |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$12.57965$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.541049041$ |
$201.2992047$ |
3.116331176 |
\( -720868474875 a^{3} + \frac{2726844170253}{2} a^{2} + \frac{3499385683905}{2} a - 1144450579761 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 16 a^{3} + 8 a^{2} - 86 a - 106\) , \( 12 a^{3} + 2 a^{2} - 55 a - 81\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(16a^{3}+8a^{2}-86a-106\right){x}+12a^{3}+2a^{2}-55a-81$ |
| 16.1-b2 |
16.1-b |
$2$ |
$2$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.57965$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.770524520$ |
$201.2992047$ |
3.116331176 |
\( \frac{622203}{2} a^{3} - \frac{2391861}{4} a^{2} - \frac{3043377}{4} a + \frac{2003589}{4} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{2} - 4\) , \( a^{2} - a - 2\) , \( -17 a^{3} + 20 a^{2} + 78 a - 38\) , \( -523 a^{3} + 632 a^{2} + 2376 a - 1303\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-17a^{3}+20a^{2}+78a-38\right){x}-523a^{3}+632a^{2}+2376a-1303$ |
| 21.1-a1 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.128168662$ |
$87.10503032$ |
3.724468981 |
\( \frac{283422372242314}{17294403} a^{3} + \frac{566742902699713}{17294403} a^{2} - \frac{11226391012049}{5764801} a - \frac{165429125280241}{17294403} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( a^{2} - 3\) , \( a\) , \( 72 a^{3} + 44 a^{2} - 417 a - 462\) , \( 786 a^{3} + 381 a^{2} - 4500 a - 4563\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(72a^{3}+44a^{2}-417a-462\right){x}+786a^{3}+381a^{2}-4500a-4563$ |
| 21.1-a2 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$4.256337325$ |
$87.10503032$ |
3.724468981 |
\( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -13750 a^{3} + 16600 a^{2} + 62447 a - 34158\) , \( 1313070 a^{3} - 1585569 a^{2} - 5963840 a + 3262240\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-13750a^{3}+16600a^{2}+62447a-34158\right){x}+1313070a^{3}-1585569a^{2}-5963840a+3262240$ |
| 21.1-a3 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{32} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$8.512674650$ |
$5.444064395$ |
3.724468981 |
\( -\frac{8726799587448504883954432291309058}{3313283022731761938915897603} a^{3} + \frac{3512609295216958802166282509619697}{1104427674243920646305299201} a^{2} + \frac{13211649577996599292098856645855219}{1104427674243920646305299201} a - \frac{7226027092698629340082787287567899}{1104427674243920646305299201} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -13750 a^{3} + 16605 a^{2} + 62452 a - 34158\) , \( 1313123 a^{3} - 1585648 a^{2} - 5964098 a + 3262387\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-13750a^{3}+16605a^{2}+62452a-34158\right){x}+1313123a^{3}-1585648a^{2}-5964098a+3262387$ |
| 21.1-a4 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$4.256337325$ |
$5.444064395$ |
3.724468981 |
\( \frac{48823353105681873974}{7203} a^{3} + \frac{126481197886698394861}{7203} a^{2} + \frac{11571783696151730607}{2401} a - \frac{18846846758411390426}{2401} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -930 a^{3} + 1070 a^{2} + 4157 a - 2268\) , \( 17576 a^{3} - 21717 a^{2} - 80504 a + 44128\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-930a^{3}+1070a^{2}+4157a-2268\right){x}+17576a^{3}-21717a^{2}-80504a+44128$ |
| 21.1-a5 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.064084331$ |
$87.10503032$ |
3.724468981 |
\( -\frac{25780022}{21609} a^{3} - \frac{18946955}{21609} a^{2} + \frac{5547721}{2401} a - \frac{16556678}{21609} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -50 a^{3} + 60 a^{2} + 227 a - 123\) , \( 382 a^{3} - 462 a^{2} - 1736 a + 949\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-50a^{3}+60a^{2}+227a-123\right){x}+382a^{3}-462a^{2}-1736a+949$ |
| 21.1-a6 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.128168662$ |
$87.10503032$ |
3.724468981 |
\( \frac{73204218707003465754418530011323778}{17294403} a^{3} - \frac{29465194712745950920219395499999665}{5764801} a^{2} - \frac{110828615043320066217968092545904435}{5764801} a + \frac{60623589840441817561364743725976875}{5764801} \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 3\) , \( -784 a^{3} - 509 a^{2} + 1519 a - 616\) , \( 18547 a^{3} + 24195 a^{2} - 19665 a + 1670\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-784a^{3}-509a^{2}+1519a-616\right){x}+18547a^{3}+24195a^{2}-19665a+1670$ |
| 21.1-b1 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.661230424$ |
$168.9045292$ |
2.243929483 |
\( \frac{2038984884625177049}{189} a^{3} - \frac{7386347250266428568}{567} a^{2} - \frac{9260854421312515172}{189} a + \frac{15197146687274038624}{567} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 54 a^{3} + 37 a^{2} - 318 a - 386\) , \( -269 a^{3} - 197 a^{2} + 1611 a + 1838\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(54a^{3}+37a^{2}-318a-386\right){x}-269a^{3}-197a^{2}+1611a+1838$ |
| 21.1-b2 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{2} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.322460848$ |
$168.9045292$ |
2.243929483 |
\( -\frac{2514435768239}{147} a^{3} + \frac{14274453830897}{441} a^{2} + \frac{2028661440026}{49} a - \frac{11951649907706}{441} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 34 a^{3} + 12 a^{2} - 188 a - 186\) , \( 183 a^{3} + 62 a^{2} - 1009 a - 987\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(34a^{3}+12a^{2}-188a-186\right){x}+183a^{3}+62a^{2}-1009a-987$ |
| 21.1-b3 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.661230424$ |
$168.9045292$ |
2.243929483 |
\( \frac{108278519}{2401} a^{3} - \frac{629396917}{7203} a^{2} - \frac{246620209}{2401} a + \frac{494923985}{7203} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + 2 a^{2} + 2 a - 1\) , \( 9 a^{3} + 3 a^{2} - 50 a - 49\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-a^{3}+2a^{2}+2a-1\right){x}+9a^{3}+3a^{2}-50a-49$ |
| 21.1-b4 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$2.644921697$ |
$10.55653307$ |
2.243929483 |
\( -\frac{15112197778035174188315}{7} a^{3} + \frac{85702869193556526085960}{21} a^{2} + \frac{36669851114797762306924}{7} a - \frac{71948817390117157608464}{21} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 0\) , \( -90 a^{3} + 80 a^{2} + 433 a - 231\) , \( 813 a^{3} - 1180 a^{2} - 3693 a + 2070\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-90a^{3}+80a^{2}+433a-231\right){x}+813a^{3}-1180a^{2}-3693a+2070$ |
| 21.1-c1 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$2.049678336$ |
$17.69737756$ |
2.915209027 |
\( \frac{2038984884625177049}{189} a^{3} - \frac{7386347250266428568}{567} a^{2} - \frac{9260854421312515172}{189} a + \frac{15197146687274038624}{567} \) |
\( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{3} - 5 a - 2\) , \( 247 a^{3} - 369 a^{2} - 783 a + 29\) , \( -2748 a^{3} + 5314 a^{2} + 6231 a - 5064\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(247a^{3}-369a^{2}-783a+29\right){x}-2748a^{3}+5314a^{2}+6231a-5064$ |
| 21.1-c2 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{2} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.024839168$ |
$283.1580409$ |
2.915209027 |
\( -\frac{2514435768239}{147} a^{3} + \frac{14274453830897}{441} a^{2} + \frac{2028661440026}{49} a - \frac{11951649907706}{441} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a + 1\) , \( 203 a^{3} + 99 a^{2} - 1170 a - 1195\) , \( -3273 a^{3} - 1665 a^{2} + 18781 a + 19343\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(203a^{3}+99a^{2}-1170a-1195\right){x}-3273a^{3}-1665a^{2}+18781a+19343$ |
| 21.1-c3 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.512419584$ |
$283.1580409$ |
2.915209027 |
\( \frac{108278519}{2401} a^{3} - \frac{629396917}{7203} a^{2} - \frac{246620209}{2401} a + \frac{494923985}{7203} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a + 1\) , \( 3 a^{3} - a^{2} - 15 a - 5\) , \( -135 a^{3} - 70 a^{2} + 775 a + 803\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(3a^{3}-a^{2}-15a-5\right){x}-135a^{3}-70a^{2}+775a+803$ |
| 21.1-c4 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.512419584$ |
$283.1580409$ |
2.915209027 |
\( -\frac{15112197778035174188315}{7} a^{3} + \frac{85702869193556526085960}{21} a^{2} + \frac{36669851114797762306924}{7} a - \frac{71948817390117157608464}{21} \) |
\( \bigl[a + 1\) , \( a^{3} - 4 a - 2\) , \( a^{2} - 2\) , \( -167 a^{3} - 178 a^{2} + 229 a - 60\) , \( 1329 a^{3} + 5176 a^{2} + 3359 a - 3168\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(-167a^{3}-178a^{2}+229a-60\right){x}+1329a^{3}+5176a^{2}+3359a-3168$ |
| 21.1-d1 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.654559453$ |
$353.3517645$ |
2.936599577 |
\( \frac{283422372242314}{17294403} a^{3} + \frac{566742902699713}{17294403} a^{2} - \frac{11226391012049}{5764801} a - \frac{165429125280241}{17294403} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 87 a^{3} - 17 a^{2} - 418 a - 327\) , \( -740 a^{3} - 97 a^{2} + 3864 a + 3503\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-5a-3\right){x}^{2}+\left(87a^{3}-17a^{2}-418a-327\right){x}-740a^{3}-97a^{2}+3864a+3503$ |
| 21.1-d2 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$6.618237813$ |
$1.380280330$ |
2.936599577 |
\( \frac{73204218707003465754418530011323778}{17294403} a^{3} - \frac{29465194712745950920219395499999665}{5764801} a^{2} - \frac{110828615043320066217968092545904435}{5764801} a + \frac{60623589840441817561364743725976875}{5764801} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -544 a^{3} + 506 a^{2} + 2629 a - 1404\) , \( -13092 a^{3} + 14266 a^{2} + 61674 a - 33259\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-544a^{3}+506a^{2}+2629a-1404\right){x}-13092a^{3}+14266a^{2}+61674a-33259$ |
| 21.1-d3 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{32} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$6.618237813$ |
$1.380280330$ |
2.936599577 |
\( -\frac{8726799587448504883954432291309058}{3313283022731761938915897603} a^{3} + \frac{3512609295216958802166282509619697}{1104427674243920646305299201} a^{2} + \frac{13211649577996599292098856645855219}{1104427674243920646305299201} a - \frac{7226027092698629340082787287567899}{1104427674243920646305299201} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -84 a^{3} + 146 a^{2} + 299 a - 174\) , \( -188 a^{3} + 858 a^{2} + 1498 a - 961\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-84a^{3}+146a^{2}+299a-174\right){x}-188a^{3}+858a^{2}+1498a-961$ |
| 21.1-d4 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$3.309118906$ |
$22.08448528$ |
2.936599577 |
\( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -34 a^{3} + 6 a^{2} + 144 a - 69\) , \( -240 a^{3} + 410 a^{2} + 1244 a - 706\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-34a^{3}+6a^{2}+144a-69\right){x}-240a^{3}+410a^{2}+1244a-706$ |
| 21.1-d5 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.827279726$ |
$353.3517645$ |
2.936599577 |
\( -\frac{25780022}{21609} a^{3} - \frac{18946955}{21609} a^{2} + \frac{5547721}{2401} a - \frac{16556678}{21609} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( a^{3} - 4 a^{2} - 6 a + 6\) , \( a^{2} - a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(a^{3}-4a^{2}-6a+6\right){x}+a^{2}-a-1$ |
| 21.1-d6 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.827279726$ |
$353.3517645$ |
2.936599577 |
\( \frac{48823353105681873974}{7203} a^{3} + \frac{126481197886698394861}{7203} a^{2} + \frac{11571783696151730607}{2401} a - \frac{18846846758411390426}{2401} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( 36 a^{3} - 554 a^{2} - 426 a + 351\) , \( 660 a^{3} + 9720 a^{2} + 6098 a - 5674\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(36a^{3}-554a^{2}-426a+351\right){x}+660a^{3}+9720a^{2}+6098a-5674$ |
| 25.1-a1 |
25.1-a |
$1$ |
$1$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5Nn |
$1$ |
\( 1 \) |
$1$ |
$260.0488896$ |
2.612402493 |
\( 256753581255500 a^{3} + 130257445927750 a^{2} - 1474438598926750 a - 1518279740390875 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a + 1\) , \( a\) , \( -4 a^{3} - a^{2} + 2 a - 2\) , \( -17 a^{3} - 13 a^{2} + 14 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a^{3}-a^{2}+2a-2\right){x}-17a^{3}-13a^{2}+14a-1$ |
| 25.1-b1 |
25.1-b |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.070938334$ |
$2779.685780$ |
2.641195305 |
\( 4036500 a^{3} - 7580142 a^{2} - 9865638 a + 6254145 \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 4 a - 1\) , \( a^{2} - a - 2\) , \( 5 a^{3} - 2 a^{2} - 20 a - 6\) , \( -a^{3} - a^{2} + 7 a + 12\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(5a^{3}-2a^{2}-20a-6\right){x}-a^{3}-a^{2}+7a+12$ |
| 25.1-b2 |
25.1-b |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.212815003$ |
$308.8539756$ |
2.641195305 |
\( 906201 a^{3} + 1751706 a^{2} - 179496 a - 477441 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a - 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 2 a^{3} - a\) , \( -58 a^{3} + 114 a^{2} + 145 a - 95\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a^{3}-a\right){x}-58a^{3}+114a^{2}+145a-95$ |
| 25.1-c1 |
25.1-c |
$1$ |
$1$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5Nn |
$4$ |
\( 1 \) |
$1$ |
$73.05431961$ |
2.935560108 |
\( 256753581255500 a^{3} + 130257445927750 a^{2} - 1474438598926750 a - 1518279740390875 \) |
\( \bigl[1\) , \( -a + 1\) , \( a^{2} - a - 2\) , \( -360 a^{3} + 434 a^{2} + 1633 a - 894\) , \( 3368 a^{3} - 4065 a^{2} - 15295 a + 8363\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-360a^{3}+434a^{2}+1633a-894\right){x}+3368a^{3}-4065a^{2}-15295a+8363$ |
| 25.1-d1 |
25.1-d |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.280250781$ |
$769.6351248$ |
2.889053175 |
\( 906201 a^{3} + 1751706 a^{2} - 179496 a - 477441 \) |
\( \bigl[a^{3} - 4 a - 2\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 3\) , \( -5 a^{3} + 3 a^{2} + 25 a + 4\) , \( 5 a^{3} - 10 a^{2} - 19 a + 31\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-5a^{3}+3a^{2}+25a+4\right){x}+5a^{3}-10a^{2}-19a+31$ |
| 25.1-d2 |
25.1-d |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.840752345$ |
$85.51501386$ |
2.889053175 |
\( 4036500 a^{3} - 7580142 a^{2} - 9865638 a + 6254145 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 2 a^{3} + 4 a^{2} - 6 a - 13\) , \( 2 a^{3} + 7 a^{2} - 6 a - 23\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(2a^{3}+4a^{2}-6a-13\right){x}+2a^{3}+7a^{2}-6a-23$ |
| 25.1-e1 |
25.1-e |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.446692615$ |
$89.94453277$ |
1.614464947 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a\) , \( a + 1\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( -5 a^{3} + 6 a^{2} + 21 a - 12\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-a^{3}+2a^{2}+3a-1\right){x}-5a^{3}+6a^{2}+21a-12$ |
| 25.1-e2 |
25.1-e |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.148897538$ |
$809.5007950$ |
1.614464947 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a\) , \( a^{3} - 5 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( -2 a^{3} + 4 a^{2} + 5 a - 6\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-a^{3}+2a^{2}+3a-1\right){x}-2a^{3}+4a^{2}+5a-6$ |
*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.
