| 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 |
| 5.2-a1 |
5.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{10} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 5 \) |
$0.156854672$ |
$1024.675807$ |
3.044649309 |
\( -\frac{1311223770134}{9765625} a^{3} + \frac{916149833587}{1953125} a^{2} - \frac{56579150647}{1953125} a - \frac{3495750181347}{9765625} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - 2 a - 3\) , \( a^{2} - a - 2\) , \( 79 a^{3} - 74 a^{2} - 466 a - 279\) , \( -65 a^{3} + 60 a^{2} + 384 a + 230\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(79a^{3}-74a^{2}-466a-279\right){x}-65a^{3}+60a^{2}+384a+230$ |
| 5.2-a2 |
5.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{5} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.078427336$ |
$1024.675807$ |
3.044649309 |
\( -\frac{494379909034744}{3125} a^{3} + \frac{345369865139817}{625} a^{2} - \frac{21243161666152}{625} a - \frac{1324563200159227}{3125} \) |
\( \bigl[a + 1\) , \( -a^{3} + 3 a^{2} + a - 5\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 19 a^{2} - 34 a - 91\) , \( -64 a^{3} + 93 a^{2} + 323 a + 62\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+a-5\right){x}^{2}+\left(19a^{2}-34a-91\right){x}-64a^{3}+93a^{2}+323a+62$ |
| 5.2-a3 |
5.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{5} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.313709345$ |
$1024.675807$ |
3.044649309 |
\( \frac{505565994}{3125} a^{3} - \frac{280031692}{625} a^{2} - \frac{290083223}{625} a + \frac{2636234227}{3125} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 5\) , \( -a^{3} + 2 a^{2} + 5 a - 1\) , \( a^{3} - 3 a^{2} - a + 4\) , \( -2 a^{3} + 3 a^{2} + 12 a\) , \( -a^{3} + 9 a + 5\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+5\right){x}{y}+\left(a^{3}-3a^{2}-a+4\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-1\right){x}^{2}+\left(-2a^{3}+3a^{2}+12a\right){x}-a^{3}+9a+5$ |
| 5.2-a4 |
5.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{20} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.313709345$ |
$64.04223793$ |
3.044649309 |
\( -\frac{2403554820306312614}{95367431640625} a^{3} + \frac{505890139014981727}{19073486328125} a^{2} + \frac{3037922838175543638}{19073486328125} a + \frac{9234734593080280813}{95367431640625} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 5\) , \( a^{3} - 3 a^{2} - 3 a + 3\) , \( a^{3} - 3 a^{2} - a + 4\) , \( 75 a^{3} - 149 a^{2} - 257 a - 65\) , \( 254 a^{3} - 68 a^{2} - 1905 a - 1296\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+5\right){x}{y}+\left(a^{3}-3a^{2}-a+4\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+3\right){x}^{2}+\left(75a^{3}-149a^{2}-257a-65\right){x}+254a^{3}-68a^{2}-1905a-1296$ |
| 5.2-b1 |
5.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{5} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.244126774$ |
$226.5802785$ |
2.095660112 |
\( -\frac{494379909034744}{3125} a^{3} + \frac{345369865139817}{625} a^{2} - \frac{21243161666152}{625} a - \frac{1324563200159227}{3125} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 5 a\) , \( a^{3} - 3 a^{2} - 2 a + 4\) , \( -32 a^{3} + 88 a^{2} + 96 a - 172\) , \( -216 a^{3} + 599 a^{2} + 616 a - 1121\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-3a^{2}-2a+4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a\right){x}^{2}+\left(-32a^{3}+88a^{2}+96a-172\right){x}-216a^{3}+599a^{2}+616a-1121$ |
| 5.2-b2 |
5.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{20} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.061031693$ |
$226.5802785$ |
2.095660112 |
\( -\frac{2403554820306312614}{95367431640625} a^{3} + \frac{505890139014981727}{19073486328125} a^{2} + \frac{3037922838175543638}{19073486328125} a + \frac{9234734593080280813}{95367431640625} \) |
\( \bigl[1\) , \( -a^{3} + 4 a^{2} - 8\) , \( a^{2} - a - 2\) , \( 183 a^{3} - 638 a^{2} + 32 a + 499\) , \( -802 a^{3} + 2797 a^{2} - 160 a - 2150\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+4a^{2}-8\right){x}^{2}+\left(183a^{3}-638a^{2}+32a+499\right){x}-802a^{3}+2797a^{2}-160a-2150$ |
| 5.2-b3 |
5.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{10} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 5 \) |
$0.122063387$ |
$906.3211142$ |
2.095660112 |
\( -\frac{1311223770134}{9765625} a^{3} + \frac{916149833587}{1953125} a^{2} - \frac{56579150647}{1953125} a - \frac{3495750181347}{9765625} \) |
\( \bigl[a + 1\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 9 a^{3} - 32 a^{2} + 25\) , \( -25 a^{3} + 89 a^{2} - 10 a - 74\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(9a^{3}-32a^{2}+25\right){x}-25a^{3}+89a^{2}-10a-74$ |
| 5.2-b4 |
5.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{5} \) |
$14.42104$ |
$(a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.061031693$ |
$906.3211142$ |
2.095660112 |
\( \frac{505565994}{3125} a^{3} - \frac{280031692}{625} a^{2} - \frac{290083223}{625} a + \frac{2636234227}{3125} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 5\) , \( -a^{2} + 3 a + 3\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( -2 a^{3} + 5 a^{2} + 11 a - 5\) , \( -a^{3} + 3 a^{2} + 6 a - 2\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+5\right){x}{y}+\left(a^{3}-3a^{2}-2a+5\right){y}={x}^{3}+\left(-a^{2}+3a+3\right){x}^{2}+\left(-2a^{3}+5a^{2}+11a-5\right){x}-a^{3}+3a^{2}+6a-2$ |
| 8.1-a1 |
8.1-a |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{15} \) |
$15.29366$ |
$(a^3-4a^2+a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$206.0884519$ |
1.561589862 |
\( \frac{47657}{32} a^{3} + \frac{69671}{16} a^{2} - \frac{20625}{32} a - \frac{46287}{8} \) |
\( \bigl[a + 1\) , \( a^{3} - 4 a^{2} + a + 6\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 12 a^{3} - 44 a^{2} + 9 a + 40\) , \( 289 a^{3} - 1009 a^{2} + 62 a + 773\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-4a^{2}+a+6\right){x}^{2}+\left(12a^{3}-44a^{2}+9a+40\right){x}+289a^{3}-1009a^{2}+62a+773$ |
| 8.1-b1 |
8.1-b |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{15} \) |
$15.29366$ |
$(a^3-4a^2+a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.049589967$ |
$142.6892678$ |
1.072329995 |
\( \frac{47657}{32} a^{3} + \frac{69671}{16} a^{2} - \frac{20625}{32} a - \frac{46287}{8} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{3} + 3 a^{2} + 2 a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( -a^{3} - 4 a^{2} + 18 a + 18\) , \( 13 a^{3} - 74 a^{2} + 72 a + 104\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-2a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-3\right){x}^{2}+\left(-a^{3}-4a^{2}+18a+18\right){x}+13a^{3}-74a^{2}+72a+104$ |
| 10.1-a1 |
10.1-a |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{11} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.123183880$ |
$682.1472050$ |
2.546861334 |
\( \frac{61368011}{10240} a^{3} - \frac{64068437}{5120} a^{2} - \frac{211051807}{10240} a - \frac{60632283}{10240} \) |
\( \bigl[1\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( a\) , \( 1\) , \( a^{3} - 3 a^{2} - 3 a + 5\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-5a+1\right){x}^{2}+{x}+a^{3}-3a^{2}-3a+5$ |
| 10.1-b1 |
10.1-b |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{11} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 11 \) |
$0.015488072$ |
$798.2193359$ |
4.121787811 |
\( \frac{61368011}{10240} a^{3} - \frac{64068437}{5120} a^{2} - \frac{211051807}{10240} a - \frac{60632283}{10240} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( a + 1\) , \( -2 a^{3} + 4 a^{2} + 7 a - 5\) , \( 6 a^{3} - 17 a^{2} - 18 a + 31\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+1\right){x}^{2}+\left(-2a^{3}+4a^{2}+7a-5\right){x}+6a^{3}-17a^{2}-18a+31$ |
| 10.2-a1 |
10.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.751900758$ |
$361.6047561$ |
2.060193350 |
\( \frac{1980843644451}{6400} a^{3} + \frac{243389866471}{640} a^{2} - \frac{469335558507}{1280} a - \frac{2336772902067}{6400} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( -a^{2} + 3 a + 2\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( 102 a^{3} - 93 a^{2} - 607 a - 364\) , \( -1005 a^{3} + 900 a^{2} + 6018 a + 3632\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+\left(a^{3}-3a^{2}-2a+5\right){y}={x}^{3}+\left(-a^{2}+3a+2\right){x}^{2}+\left(102a^{3}-93a^{2}-607a-364\right){x}-1005a^{3}+900a^{2}+6018a+3632$ |
| 10.2-a2 |
10.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5^{4} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.503801517$ |
$22.60029726$ |
2.060193350 |
\( \frac{38452034491075683346111}{10000} a^{3} + \frac{4444544130979740843651}{1000} a^{2} - \frac{10399253362696020625687}{2000} a - \frac{48737203898327578291887}{10000} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( -a^{2} + 3 a + 2\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( 122 a^{3} - 158 a^{2} - 617 a - 324\) , \( -865 a^{3} + 584 a^{2} + 5630 a + 3584\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+\left(a^{3}-3a^{2}-2a+5\right){y}={x}^{3}+\left(-a^{2}+3a+2\right){x}^{2}+\left(122a^{3}-158a^{2}-617a-324\right){x}-865a^{3}+584a^{2}+5630a+3584$ |
| 10.2-a3 |
10.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.503801517$ |
$361.6047561$ |
2.060193350 |
\( \frac{6191488811349}{80} a^{3} - \frac{1713890395399}{8} a^{2} - \frac{3558476086237}{16} a + \frac{32241492530587}{80} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -1\) , \( a^{3} - 3 a^{2} - a + 4\) , \( -4 a^{3} - 4 a^{2} - 6\) , \( -23 a^{3} - 26 a^{2} + 40 a + 36\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-3a^{2}-a+4\right){y}={x}^{3}-{x}^{2}+\left(-4a^{3}-4a^{2}-6\right){x}-23a^{3}-26a^{2}+40a+36$ |
| 10.2-a4 |
10.2-a |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.375950379$ |
$361.6047561$ |
2.060193350 |
\( -\frac{412041851931236474929}{327680} a^{3} + \frac{36868201279503059363}{32768} a^{2} + \frac{493537656936056819913}{65536} a + \frac{1491241609966012887393}{327680} \) |
\( \bigl[a + 1\) , \( -a^{3} + 3 a^{2} + 2 a - 5\) , \( a\) , \( -39 a^{3} + 45 a^{2} + 87 a - 100\) , \( -114 a^{3} - 362 a^{2} + 65 a + 524\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-5\right){x}^{2}+\left(-39a^{3}+45a^{2}+87a-100\right){x}-114a^{3}-362a^{2}+65a+524$ |
| 10.2-b1 |
10.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{4} \cdot 5^{2} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$1135.450290$ |
4.301812271 |
\( -\frac{1768504989}{400} a^{3} + \frac{158379991}{40} a^{2} + \frac{2119007653}{80} a + \frac{6403564813}{400} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 116 a^{3} - 403 a^{2} + 20 a + 309\) , \( 572 a^{3} - 1998 a^{2} + 123 a + 1531\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(116a^{3}-403a^{2}+20a+309\right){x}+572a^{3}-1998a^{2}+123a+1531$ |
| 10.2-b2 |
10.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1135.450290$ |
4.301812271 |
\( \frac{132261}{20} a^{3} - \frac{34149}{2} a^{2} - \frac{85437}{4} a + \frac{592123}{20} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{2} - 3 a - 4\) , \( a^{3} - 3 a^{2} - a + 4\) , \( -a^{3} + 3 a^{2} + 3 a - 4\) , \( -a^{2} + a\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-3a^{2}-a+4\right){y}={x}^{3}+\left(a^{2}-3a-4\right){x}^{2}+\left(-a^{3}+3a^{2}+3a-4\right){x}-a^{2}+a$ |
| 10.2-b3 |
10.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{8} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$283.8625726$ |
4.301812271 |
\( \frac{169055205071}{1280} a^{3} + \frac{19164023843}{128} a^{2} - \frac{46019151671}{256} a - \frac{208015302047}{1280} \) |
\( \bigl[1\) , \( -a - 1\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( -19 a^{3} + 52 a^{2} + 56 a - 99\) , \( -74 a^{3} + 205 a^{2} + 212 a - 385\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-3a^{2}-2a+5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-19a^{3}+52a^{2}+56a-99\right){x}-74a^{3}+205a^{2}+212a-385$ |
| 10.2-b4 |
10.2-b |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5^{4} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$283.8625726$ |
4.301812271 |
\( -\frac{488238516060573841}{2500} a^{3} + \frac{43686037809016669}{250} a^{2} + \frac{584804897662713597}{500} a + \frac{1767008827760843297}{2500} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 5\) , \( -1\) , \( a + 1\) , \( -69 a^{3} - 142 a^{2} + 68 a + 189\) , \( 694 a^{3} + 941 a^{2} - 885 a - 1109\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+5\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-69a^{3}-142a^{2}+68a+189\right){x}+694a^{3}+941a^{2}-885a-1109$ |
| 10.2-c1 |
10.2-c |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{4} \cdot 5^{2} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.352071729$ |
$719.6915421$ |
1.919954231 |
\( -\frac{1768504989}{400} a^{3} + \frac{158379991}{40} a^{2} + \frac{2119007653}{80} a + \frac{6403564813}{400} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{3} + 4 a^{2} - 7\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( 23 a^{3} - 65 a^{2} - 31 a + 23\) , \( 87 a^{3} - 258 a^{2} - 90 a + 119\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+4a^{2}-7\right){x}^{2}+\left(23a^{3}-65a^{2}-31a+23\right){x}+87a^{3}-258a^{2}-90a+119$ |
| 10.2-c2 |
10.2-c |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.704143458$ |
$719.6915421$ |
1.919954231 |
\( \frac{132261}{20} a^{3} - \frac{34149}{2} a^{2} - \frac{85437}{4} a + \frac{592123}{20} \) |
\( \bigl[a + 1\) , \( -a^{3} + 4 a^{2} - a - 6\) , \( a + 1\) , \( -2 a^{3} + 7 a^{2} - 2 a - 5\) , \( -2 a^{3} + 7 a^{2} - a - 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+4a^{2}-a-6\right){x}^{2}+\left(-2a^{3}+7a^{2}-2a-5\right){x}-2a^{3}+7a^{2}-a-6$ |
| 10.2-c3 |
10.2-c |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5^{4} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.704143458$ |
$44.98072138$ |
1.919954231 |
\( -\frac{488238516060573841}{2500} a^{3} + \frac{43686037809016669}{250} a^{2} + \frac{584804897662713597}{500} a + \frac{1767008827760843297}{2500} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 5\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - 3 a^{2} - a + 5\) , \( -7 a^{3} - 39 a^{2} - 29 a + 17\) , \( -45 a^{3} - 97 a^{2} - 88 a - 68\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+5\right){x}{y}+\left(a^{3}-3a^{2}-a+5\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(-7a^{3}-39a^{2}-29a+17\right){x}-45a^{3}-97a^{2}-88a-68$ |
| 10.2-c4 |
10.2-c |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{8} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.176035864$ |
$719.6915421$ |
1.919954231 |
\( \frac{169055205071}{1280} a^{3} + \frac{19164023843}{128} a^{2} - \frac{46019151671}{256} a - \frac{208015302047}{1280} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 5\) , \( a^{3} - 4 a^{2} + 7\) , \( 0\) , \( -2 a^{3} + 7 a^{2} + 3 a - 17\) , \( -22 a^{3} + 67 a^{2} + 53 a - 143\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+5\right){x}{y}={x}^{3}+\left(a^{3}-4a^{2}+7\right){x}^{2}+\left(-2a^{3}+7a^{2}+3a-17\right){x}-22a^{3}+67a^{2}+53a-143$ |
| 10.2-d1 |
10.2-d |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$452.7817171$ |
3.430853756 |
\( \frac{6191488811349}{80} a^{3} - \frac{1713890395399}{8} a^{2} - \frac{3558476086237}{16} a + \frac{32241492530587}{80} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( a^{3} - 2 a^{2} - 3 a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -23 a^{3} - 19 a^{2} + 31 a + 15\) , \( 221 a^{3} + 248 a^{2} - 302 a - 269\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a\right){x}^{2}+\left(-23a^{3}-19a^{2}+31a+15\right){x}+221a^{3}+248a^{2}-302a-269$ |
| 10.2-d2 |
10.2-d |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5 \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$113.1954292$ |
3.430853756 |
\( -\frac{412041851931236474929}{327680} a^{3} + \frac{36868201279503059363}{32768} a^{2} + \frac{493537656936056819913}{65536} a + \frac{1491241609966012887393}{327680} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - 3 a - 3\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( -7 a^{3} + 14 a^{2} - 49\) , \( 23 a^{3} + 4 a^{2} - 68 a - 33\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-3a^{2}-2a+5\right){y}={x}^{3}+\left(a^{2}-3a-3\right){x}^{2}+\left(-7a^{3}+14a^{2}-49\right){x}+23a^{3}+4a^{2}-68a-33$ |
| 10.2-d3 |
10.2-d |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$452.7817171$ |
3.430853756 |
\( \frac{1980843644451}{6400} a^{3} + \frac{243389866471}{640} a^{2} - \frac{469335558507}{1280} a - \frac{2336772902067}{6400} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 5\) , \( -a^{2} + 3 a + 3\) , \( a^{2} - a - 3\) , \( 9 a^{3} - 30 a^{2} - 4 a + 27\) , \( 18 a^{3} - 55 a^{2} - 16 a + 29\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+5\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+3a+3\right){x}^{2}+\left(9a^{3}-30a^{2}-4a+27\right){x}+18a^{3}-55a^{2}-16a+29$ |
| 10.2-d4 |
10.2-d |
$4$ |
$4$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5^{4} \) |
$15.72625$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$113.1954292$ |
3.430853756 |
\( \frac{38452034491075683346111}{10000} a^{3} + \frac{4444544130979740843651}{1000} a^{2} - \frac{10399253362696020625687}{2000} a - \frac{48737203898327578291887}{10000} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 5\) , \( -a^{2} + 3 a + 3\) , \( a^{2} - a - 3\) , \( 119 a^{3} - 420 a^{2} + 21 a + 327\) , \( 1374 a^{3} - 4779 a^{2} + 270 a + 3653\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+5\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+3a+3\right){x}^{2}+\left(119a^{3}-420a^{2}+21a+327\right){x}+1374a^{3}-4779a^{2}+270a+3653$ |
| 16.1-a1 |
16.1-a |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{5} \) |
$16.67786$ |
$(-a^2+2), (a^3-4a^2+a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.052108857$ |
$1776.122160$ |
5.610320794 |
\( \frac{41013}{4} a^{3} - 7323 a^{2} - \frac{256571}{4} a - \frac{180489}{4} \) |
\( \bigl[1\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( 0\) , \( 2 a^{3} - 3 a^{2} - 10 a - 1\) , \( -4 a^{3} + 3 a^{2} + 25 a + 17\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(2a^{3}-3a^{2}-10a-1\right){x}-4a^{3}+3a^{2}+25a+17$ |
| 16.1-b1 |
16.1-b |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{5} \) |
$16.67786$ |
$(-a^2+2), (a^3-4a^2+a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.076788529$ |
$503.7360573$ |
2.344783223 |
\( \frac{41013}{4} a^{3} - 7323 a^{2} - \frac{256571}{4} a - \frac{180489}{4} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 3 a^{2} - a + 5\) , \( a^{3} - 3 a^{2} - a + 4\) , \( a^{2} - a + 5\) , \( 2 a^{3} - 6 a^{2} + a + 5\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-3a^{2}-a+4\right){y}={x}^{3}+\left(a^{3}-3a^{2}-a+5\right){x}^{2}+\left(a^{2}-a+5\right){x}+2a^{3}-6a^{2}+a+5$ |
| 17.1-a1 |
17.1-a |
$2$ |
$3$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$16.80472$ |
$(a^2-3a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$36$ |
\( 1 \) |
$1$ |
$8.475511894$ |
2.311967701 |
\( -\frac{2256615247347497592}{4913} a^{3} - \frac{2608347573300636097}{4913} a^{2} + \frac{3051478707122955964}{4913} a + \frac{2860216264558380576}{4913} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 4\) , \( -a^{3} + 4 a^{2} + a - 6\) , \( a^{2} - 2 a - 3\) , \( -78 a^{3} + 206 a^{2} + 212 a - 382\) , \( -548 a^{3} + 1439 a^{2} + 1538 a - 2722\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+4\right){x}{y}+\left(a^{2}-2a-3\right){y}={x}^{3}+\left(-a^{3}+4a^{2}+a-6\right){x}^{2}+\left(-78a^{3}+206a^{2}+212a-382\right){x}-548a^{3}+1439a^{2}+1538a-2722$ |
| 17.1-a2 |
17.1-a |
$2$ |
$3$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$16.80472$ |
$(a^2-3a-1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$4$ |
\( 1 \) |
$1$ |
$686.5164634$ |
2.311967701 |
\( \frac{3229332}{17} a^{3} - \frac{11822438}{17} a^{2} + \frac{866831}{17} a + \frac{9135204}{17} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 4\) , \( -a^{3} + 4 a^{2} + a - 6\) , \( a^{2} - 2 a - 3\) , \( -3 a^{3} + 11 a^{2} + 2 a - 12\) , \( 2 a^{2} - 5 a - 1\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+4\right){x}{y}+\left(a^{2}-2a-3\right){y}={x}^{3}+\left(-a^{3}+4a^{2}+a-6\right){x}^{2}+\left(-3a^{3}+11a^{2}+2a-12\right){x}+2a^{2}-5a-1$ |
| 17.1-b1 |
17.1-b |
$2$ |
$3$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$16.80472$ |
$(a^2-3a-1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.122384328$ |
$236.4168761$ |
3.507822348 |
\( -\frac{2256615247347497592}{4913} a^{3} - \frac{2608347573300636097}{4913} a^{2} + \frac{3051478707122955964}{4913} a + \frac{2860216264558380576}{4913} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 3 a^{2} - a + 4\) , \( a^{2} - a - 3\) , \( -15 a^{3} + 55 a^{2} + 15 a - 144\) , \( -173 a^{3} + 405 a^{2} + 621 a - 554\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-a+4\right){x}^{2}+\left(-15a^{3}+55a^{2}+15a-144\right){x}-173a^{3}+405a^{2}+621a-554$ |
| 17.1-b2 |
17.1-b |
$2$ |
$3$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$16.80472$ |
$(a^2-3a-1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.013598258$ |
$2127.751885$ |
3.507822348 |
\( \frac{3229332}{17} a^{3} - \frac{11822438}{17} a^{2} + \frac{866831}{17} a + \frac{9135204}{17} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 3 a^{2} - a + 4\) , \( a^{2} - a - 3\) , \( 6\) , \( a^{3} - a^{2} - 4 a - 3\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-a+4\right){x}^{2}+6{x}+a^{3}-a^{2}-4a-3$ |
| 20.1-a1 |
20.1-a |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$17.14960$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$23.28379615$ |
1.058567028 |
\( -\frac{26705567264078}{25} a^{3} + \frac{14784945779449}{5} a^{2} + \frac{15348670107856}{5} a - \frac{139066281441424}{25} \) |
\( \bigl[a^{2} - 2 a - 2\) , \( a^{3} - 4 a^{2} + 8\) , \( a^{3} - 3 a^{2} - 2 a + 4\) , \( 5 a^{3} - 12 a^{2} - 14 a + 7\) , \( 35 a^{3} - 10 a^{2} - 261 a - 177\bigr] \) |
${y}^2+\left(a^{2}-2a-2\right){x}{y}+\left(a^{3}-3a^{2}-2a+4\right){y}={x}^{3}+\left(a^{3}-4a^{2}+8\right){x}^{2}+\left(5a^{3}-12a^{2}-14a+7\right){x}+35a^{3}-10a^{2}-261a-177$ |
| 20.1-b1 |
20.1-b |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5 \) |
$17.14960$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \) |
$1$ |
$143.6704851$ |
3.265894392 |
\( \frac{5697642}{5} a^{3} - 3937265 a^{2} + 66954 a + \frac{14444936}{5} \) |
\( \bigl[a^{3} - 3 a^{2} - a + 4\) , \( a^{3} - 3 a^{2} - 3 a + 5\) , \( a^{2} - 2 a - 2\) , \( a^{3} - 4 a^{2} - 13 a + 7\) , \( -2 a^{3} - 6 a^{2} + 7 a + 18\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+4\right){x}{y}+\left(a^{2}-2a-2\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+5\right){x}^{2}+\left(a^{3}-4a^{2}-13a+7\right){x}-2a^{3}-6a^{2}+7a+18$ |
| 20.1-c1 |
20.1-c |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5 \) |
$17.14960$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$143.2382704$ |
1.085356452 |
\( \frac{5697642}{5} a^{3} - 3937265 a^{2} + 66954 a + \frac{14444936}{5} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{2} - 2 a - 2\) , \( -4 a^{3} - 8 a^{2} + 5 a + 9\) , \( 17 a^{3} + 17 a^{2} - 24 a - 20\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2a-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(-4a^{3}-8a^{2}+5a+9\right){x}+17a^{3}+17a^{2}-24a-20$ |
| 20.1-d1 |
20.1-d |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$17.14960$ |
$(-a^2+2), (a^2+a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$131.5004063$ |
1.992830744 |
\( -\frac{26705567264078}{25} a^{3} + \frac{14784945779449}{5} a^{2} + \frac{15348670107856}{5} a - \frac{139066281441424}{25} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 3 a^{2} + a - 4\) , \( a\) , \( 14 a^{3} - 50 a^{2} + 4 a + 39\) , \( -90 a^{3} + 314 a^{2} - 19 a - 241\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a^{2}+a-4\right){x}^{2}+\left(14a^{3}-50a^{2}+4a+39\right){x}-90a^{3}+314a^{2}-19a-241$ |
| 25.3-a1 |
25.3-a |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$17.63469$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$25$ |
\( 1 \) |
$1$ |
$20.68274559$ |
3.917973757 |
\( 2898095937024000 a^{3} - 8022333090816000 a^{2} - 8328210839445504 a + 15091513438031872 \) |
\( \bigl[0\) , \( -a^{3} + 4 a^{2} + a - 6\) , \( a^{2} - 2 a - 3\) , \( -471 a^{3} + 1343 a^{2} + 1301 a - 2611\) , \( -9062 a^{3} + 24975 a^{2} + 26354 a - 46475\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){y}={x}^{3}+\left(-a^{3}+4a^{2}+a-6\right){x}^{2}+\left(-471a^{3}+1343a^{2}+1301a-2611\right){x}-9062a^{3}+24975a^{2}+26354a-46475$ |
| 25.3-a2 |
25.3-a |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{2} \) |
$17.63469$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$517.0686399$ |
3.917973757 |
\( -4096 a^{3} + 8192 a^{2} - 4096 \) |
\( \bigl[0\) , \( -a^{3} + 4 a^{2} + a - 6\) , \( a^{2} - 2 a - 3\) , \( -a^{3} + 3 a^{2} + a - 1\) , \( -a^{2} - 4 a - 1\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){y}={x}^{3}+\left(-a^{3}+4a^{2}+a-6\right){x}^{2}+\left(-a^{3}+3a^{2}+a-1\right){x}-a^{2}-4a-1$ |
| 25.3-b1 |
25.3-b |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{10} \) |
$17.63469$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$297.7004919$ |
2.255759922 |
\( 8532872 a^{3} - 7635482 a^{2} - 51101467 a - 30878705 \) |
\( \bigl[a + 1\) , \( -a^{3} + 4 a^{2} - 7\) , \( a^{2} - a - 2\) , \( 2 a^{2} - 2 a + 2\) , \( 3 a^{2} - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+4a^{2}-7\right){x}^{2}+\left(2a^{2}-2a+2\right){x}+3a^{2}-3$ |
| 25.3-c1 |
25.3-c |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{10} \) |
$17.63469$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$52.66284059$ |
0.399041077 |
\( 8532872 a^{3} - 7635482 a^{2} - 51101467 a - 30878705 \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( a^{3} - 3 a^{2} - a + 4\) , \( 15 a^{3} - 12 a^{2} - 88 a - 52\) , \( 69 a^{3} - 61 a^{2} - 407 a - 247\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-3a^{2}-a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(15a^{3}-12a^{2}-88a-52\right){x}+69a^{3}-61a^{2}-407a-247$ |
| 25.3-d1 |
25.3-d |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$17.63469$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$3.888060765$ |
0.736523107 |
\( 2898095937024000 a^{3} - 8022333090816000 a^{2} - 8328210839445504 a + 15091513438031872 \) |
\( \bigl[0\) , \( -a^{2} + a + 2\) , \( 1\) , \( -1524 a^{3} + 4247 a^{2} + 4390 a - 7983\) , \( -53706 a^{3} + 148100 a^{2} + 154114 a - 278736\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-1524a^{3}+4247a^{2}+4390a-7983\right){x}-53706a^{3}+148100a^{2}+154114a-278736$ |
| 25.3-d2 |
25.3-d |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{2} \) |
$17.63469$ |
$(a^2-2a-3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2430.037978$ |
0.736523107 |
\( -4096 a^{3} + 8192 a^{2} - 4096 \) |
\( \bigl[0\) , \( -a^{2} + a + 2\) , \( 1\) , \( -4 a^{3} + 7 a^{2} + 10 a - 13\) , \( 2 a^{3} + 14 a^{2} + 2 a - 22\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-4a^{3}+7a^{2}+10a-13\right){x}+2a^{3}+14a^{2}+2a-22$ |
| 25.3-e1 |
25.3-e |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.63469$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1.441860943$ |
$189.6715439$ |
8.288937648 |
\( 2898095937024000 a^{3} - 8022333090816000 a^{2} - 8328210839445504 a + 15091513438031872 \) |
\( \bigl[0\) , \( -a^{3} + 3 a^{2} + 2 a - 3\) , \( a^{2} - a - 3\) , \( 373 a^{3} - 1296 a^{2} + 64 a + 984\) , \( 8628 a^{3} - 30147 a^{2} + 1876 a + 23139\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-3\right){x}^{2}+\left(373a^{3}-1296a^{2}+64a+984\right){x}+8628a^{3}-30147a^{2}+1876a+23139$ |
| 25.3-e2 |
25.3-e |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.63469$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.288372188$ |
$948.3577195$ |
8.288937648 |
\( -4096 a^{3} + 8192 a^{2} - 4096 \) |
\( \bigl[0\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{3} - 3 a^{2} - a + 5\) , \( -3 a^{3} + 5 a^{2} + 10 a - 6\) , \( a^{3} + a^{2} - 2 a - 5\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+5\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+1\right){x}^{2}+\left(-3a^{3}+5a^{2}+10a-6\right){x}+a^{3}+a^{2}-2a-5$ |
| 25.3-f1 |
25.3-f |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.63469$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.066744517$ |
$704.7128603$ |
4.276833699 |
\( 8532872 a^{3} - 7635482 a^{2} - 51101467 a - 30878705 \) |
\( \bigl[a^{3} - 3 a^{2} - a + 5\) , \( a^{2} - 2 a - 3\) , \( a^{3} - 3 a^{2} - a + 4\) , \( 2 a^{2} - 2 a - 6\) , \( -2 a^{3} + 3 a^{2} + 12 a + 5\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-a+5\right){x}{y}+\left(a^{3}-3a^{2}-a+4\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(2a^{2}-2a-6\right){x}-2a^{3}+3a^{2}+12a+5$ |
| 25.3-g1 |
25.3-g |
$1$ |
$1$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.63469$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.654079840$ |
$117.9181500$ |
2.337678252 |
\( 8532872 a^{3} - 7635482 a^{2} - 51101467 a - 30878705 \) |
\( \bigl[1\) , \( -a^{3} + 3 a^{2} + a - 3\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{3} + 7 a^{2} + a - 6\) , \( -a^{3} + 2 a^{2} - 2 a - 4\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+a-3\right){x}^{2}+\left(-a^{3}+7a^{2}+a-6\right){x}-a^{3}+2a^{2}-2a-4$ |
| 25.3-h1 |
25.3-h |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.63469$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 3 \) |
$0.036978346$ |
$373.0563770$ |
1.254343617 |
\( -4096 a^{3} + 8192 a^{2} - 4096 \) |
\( \bigl[0\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( a + 1\) , \( -8 a^{3} + 13 a^{2} + 19 a - 26\) , \( -9 a^{3} - 10 a^{2} + 12 a + 10\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+5\right){x}^{2}+\left(-8a^{3}+13a^{2}+19a-26\right){x}-9a^{3}-10a^{2}+12a+10$ |
| 25.3-h2 |
25.3-h |
$2$ |
$5$ |
4.4.17417.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.63469$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 3 \) |
$0.184891731$ |
$74.61127540$ |
1.254343617 |
\( 2898095937024000 a^{3} - 8022333090816000 a^{2} - 8328210839445504 a + 15091513438031872 \) |
\( \bigl[0\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( 292 a^{3} - 418 a^{2} - 1376 a - 672\) , \( 4836 a^{3} - 5586 a^{2} - 25956 a - 14452\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+5\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+5\right){x}^{2}+\left(292a^{3}-418a^{2}-1376a-672\right){x}+4836a^{3}-5586a^{2}-25956a-14452$ |
*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.
