| 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 |
| 2.1-a1 |
2.1-a |
$2$ |
$2$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
2.1 |
\( 2 \) |
\( 2^{22} \) |
$12.55854$ |
$(-a^2-a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$443.9006078$ |
1.722201651 |
\( -\frac{1090590304095}{4194304} a^{3} + \frac{2375121316619}{4194304} a^{2} + \frac{1338253439801}{2097152} a - \frac{4682915664763}{4194304} \) |
\( \bigl[a^{2} - 4\) , \( a + 1\) , \( a^{3} - 4 a\) , \( 3 a^{3} - 14 a - 7\) , \( 6 a^{3} + 10 a^{2} - 37 a - 57\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a^{3}-14a-7\right){x}+6a^{3}+10a^{2}-37a-57$ |
| 2.1-a2 |
2.1-a |
$2$ |
$2$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
2.1 |
\( 2 \) |
\( 2^{11} \) |
$12.55854$ |
$(-a^2-a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$887.8012156$ |
1.722201651 |
\( -\frac{683883591}{2048} a^{3} - \frac{814036973}{2048} a^{2} + \frac{1917728913}{1024} a + \frac{5242795261}{2048} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 4 a + 8\) , \( a + 1\) , \( 2 a^{3} - 3 a^{2} - 10 a + 11\) , \( 2 a^{3} - 8 a + 2\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+8\right){x}^{2}+\left(2a^{3}-3a^{2}-10a+11\right){x}+2a^{3}-8a+2$ |
| 3.1-a1 |
3.1-a |
$1$ |
$1$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$13.21145$ |
$(a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.103254519$ |
$496.2882948$ |
3.180983001 |
\( \frac{86056}{9} a^{3} + 10640 a^{2} - \frac{484492}{9} a - \frac{644893}{9} \) |
\( \bigl[a\) , \( a^{2} - a - 5\) , \( 0\) , \( a^{3} - 3 a^{2} - 5 a + 15\) , \( 8 a^{3} - 11 a^{2} - 39 a + 45\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(a^{3}-3a^{2}-5a+15\right){x}+8a^{3}-11a^{2}-39a+45$ |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( -5 \) |
$14.08256$ |
$(a^2-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.092709732$ |
$530.6537480$ |
1.526949985 |
\( \frac{866127}{5} a^{3} - \frac{1338841}{5} a^{2} - \frac{3957031}{5} a + \frac{5114241}{5} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 4\) , \( -a^{3} + a^{2} + 5 a - 4\) , \( a^{3} - a^{2} - 3 a + 4\) , \( 10 a^{3} + 13 a^{2} - 57 a - 87\) , \( 1401 a^{3} + 1654 a^{2} - 7853 a - 10677\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+4\right){x}{y}+\left(a^{3}-a^{2}-3a+4\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-4\right){x}^{2}+\left(10a^{3}+13a^{2}-57a-87\right){x}+1401a^{3}+1654a^{2}-7853a-10677$ |
| 6.1-a1 |
6.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{3} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$121.5398435$ |
0.943076515 |
\( -\frac{77907132001262368085}{54} a^{3} + \frac{18477124634925465181}{6} a^{2} + \frac{95196121451573274301}{27} a - \frac{328488405884345196283}{54} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{2} + 5\) , \( 1\) , \( 2137 a^{3} + 2522 a^{2} - 11978 a - 16275\) , \( -121578 a^{3} - 143596 a^{2} + 681447 a + 926435\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(2137a^{3}+2522a^{2}-11978a-16275\right){x}-121578a^{3}-143596a^{2}+681447a+926435$ |
| 6.1-a2 |
6.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{6} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$486.1593740$ |
0.943076515 |
\( -\frac{1106561590963}{2916} a^{3} + \frac{262445678183}{324} a^{2} + \frac{1352062034885}{1458} a - \frac{4665604095371}{2916} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{2} + 5\) , \( 1\) , \( 147 a^{3} + 172 a^{2} - 823 a - 1110\) , \( -1242 a^{3} - 1468 a^{2} + 6962 a + 9470\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(147a^{3}+172a^{2}-823a-1110\right){x}-1242a^{3}-1468a^{2}+6962a+9470$ |
| 6.1-a3 |
6.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{3} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$486.1593740$ |
0.943076515 |
\( \frac{12062057}{54} a^{3} + \frac{3439547}{6} a^{2} - \frac{6216655}{27} a - \frac{46903469}{54} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -a\) , \( -a^{2} - a + 2\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}-a{x}-a^{2}-a+2$ |
| 6.1-a4 |
6.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{12} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$121.5398435$ |
0.943076515 |
\( \frac{21310652377623137}{8503056} a^{3} - \frac{3485294491586413}{944784} a^{2} - \frac{51501880555322359}{4251528} a + \frac{130302930688223989}{8503056} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( a^{2} - a - 5\) , \( a^{2} + a - 4\) , \( 10 a^{3} - 20 a^{2} - 21 a + 44\) , \( 62 a^{3} - 131 a^{2} - 153 a + 257\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(10a^{3}-20a^{2}-21a+44\right){x}+62a^{3}-131a^{2}-153a+257$ |
| 6.1-b1 |
6.1-b |
$1$ |
$1$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.047632819$ |
$437.1429008$ |
2.585103631 |
\( -\frac{2337655068281}{2304} a^{3} - \frac{629960091259}{256} a^{2} + \frac{1306397721391}{1152} a + \frac{8674635035459}{2304} \) |
\( \bigl[1\) , \( -a^{3} + 5 a\) , \( a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - 7 a^{2} + 9 a + 5\) , \( -35 a^{3} + 63 a^{2} + 125 a - 175\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(a^{3}-7a^{2}+9a+5\right){x}-35a^{3}+63a^{2}+125a-175$ |
| 6.1-c1 |
6.1-c |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{4} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.074119236$ |
$942.1296777$ |
2.167354094 |
\( \frac{7871}{162} a^{3} - \frac{16789}{18} a^{2} + \frac{17528}{81} a + \frac{839641}{162} \) |
\( \bigl[a^{2} - 4\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( 8 a^{3} + 11 a^{2} - 47 a - 68\) , \( 43 a^{3} + 50 a^{2} - 241 a - 326\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(8a^{3}+11a^{2}-47a-68\right){x}+43a^{3}+50a^{2}-241a-326$ |
| 6.1-c2 |
6.1-c |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.074119236$ |
$1884.259355$ |
2.167354094 |
\( \frac{222772878257}{1296} a^{3} - \frac{45821995357}{144} a^{2} - \frac{526090042975}{648} a + \frac{1799572142389}{1296} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 3\) , \( a^{3} - 3 a - 1\) , \( 22 a^{3} - 22 a^{2} - 68 a + 19\) , \( 18 a^{3} - 58 a^{2} - 7 a + 173\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+3\right){x}^{2}+\left(22a^{3}-22a^{2}-68a+19\right){x}+18a^{3}-58a^{2}-7a+173$ |
| 6.1-c3 |
6.1-c |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{2} \cdot 3^{2} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.148238473$ |
$471.0648388$ |
2.167354094 |
\( \frac{9519910040935912187}{36} a^{3} - \frac{1556951551395488843}{4} a^{2} - \frac{23006985330981228379}{18} a + \frac{58209060555516158899}{36} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 3\) , \( a^{3} - 3 a - 1\) , \( 192 a^{3} - 387 a^{2} - 483 a + 739\) , \( 3037 a^{3} - 6509 a^{2} - 7385 a + 12917\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+3\right){x}^{2}+\left(192a^{3}-387a^{2}-483a+739\right){x}+3037a^{3}-6509a^{2}-7385a+12917$ |
| 6.1-c4 |
6.1-c |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{8} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.037059618$ |
$1884.259355$ |
2.167354094 |
\( -\frac{97259155}{26244} a^{3} + \frac{20327771}{2916} a^{2} + \frac{14950541}{13122} a + \frac{319978897}{26244} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - 2 a^{2} - 5 a + 7\) , \( 0\) , \( -a^{3} + 2 a^{2} + 5 a - 8\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-5a+7\right){x}^{2}+\left(-a^{3}+2a^{2}+5a-8\right){x}$ |
| 6.1-c5 |
6.1-c |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{16} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.074119236$ |
$235.5324194$ |
2.167354094 |
\( -\frac{50692999978511843}{86093442} a^{3} + \frac{12032819035554055}{9565938} a^{2} + \frac{61955815036631242}{43046721} a - \frac{213899426219130907}{86093442} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - 2 a^{2} - 5 a + 7\) , \( 0\) , \( 4 a^{3} - 8 a^{2} - 20 a + 32\) , \( 38 a^{3} - 61 a^{2} - 186 a + 251\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-5a+7\right){x}^{2}+\left(4a^{3}-8a^{2}-20a+32\right){x}+38a^{3}-61a^{2}-186a+251$ |
| 6.1-c6 |
6.1-c |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{8} \cdot 3^{2} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.148238473$ |
$471.0648388$ |
2.167354094 |
\( -\frac{15230657487296245223}{2304} a^{3} - \frac{1998760170579479973}{256} a^{2} + \frac{42684042818617248625}{1152} a + \frac{116058410062930187741}{2304} \) |
\( \bigl[1\) , \( -a^{2} - a + 3\) , \( 0\) , \( -a^{3} + 5 a^{2} + 4 a - 14\) , \( a^{3} - 2 a + 3\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-a^{3}+5a^{2}+4a-14\right){x}+a^{3}-2a+3$ |
| 6.1-d1 |
6.1-d |
$2$ |
$3$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{3} \) |
$0.033181101$ |
$320.4693824$ |
2.640314031 |
\( -\frac{258425}{144} a^{3} + \frac{39461}{16} a^{2} + \frac{628615}{72} a - \frac{1446733}{144} \) |
\( \bigl[a^{2} - 4\) , \( a^{3} - 4 a - 2\) , \( a^{3} - a^{2} - 4 a + 3\) , \( 4\) , \( -3\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+4{x}-3$ |
| 6.1-d2 |
6.1-d |
$2$ |
$3$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3^{6} \) |
$14.40719$ |
$(-a^2-a+2), (a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.011060367$ |
$320.4693824$ |
2.640314031 |
\( \frac{1362928582999}{2985984} a^{3} - \frac{323081055755}{331776} a^{2} - \frac{1666760035841}{1492992} a + \frac{5744608553555}{2985984} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} - a^{2} - 5 a + 2\) , \( 1\) , \( 8 a^{3} - 15 a^{2} - 26 a + 43\) , \( -11 a^{3} + 28 a^{2} + 17 a - 43\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(8a^{3}-15a^{2}-26a+43\right){x}-11a^{3}+28a^{2}+17a-43$ |
| 8.2-a1 |
8.2-a |
$2$ |
$2$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{11} \) |
$14.93470$ |
$(-a^2-a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$579.4419672$ |
1.124030802 |
\( 3077 a^{3} - 7048 a^{2} - 7962 a + 16121 \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} - 3\) , \( -8 a^{3} + 11 a^{2} + 37 a - 44\) , \( -7 a^{3} + 10 a^{2} + 33 a - 41\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-8a^{3}+11a^{2}+37a-44\right){x}-7a^{3}+10a^{2}+33a-41$ |
| 8.2-a2 |
8.2-a |
$2$ |
$2$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{10} \) |
$14.93470$ |
$(-a^2-a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$289.7209836$ |
1.124030802 |
\( -87719585 a^{3} + 187620323 a^{2} + 214483355 a - 370523389 \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 4 a + 6\) , \( a^{2} - 3\) , \( -26 a^{3} - 33 a^{2} + 144 a + 211\) , \( -74 a^{3} - 91 a^{2} + 416 a + 580\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+6\right){x}^{2}+\left(-26a^{3}-33a^{2}+144a+211\right){x}-74a^{3}-91a^{2}+416a+580$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$15.15621$ |
$(a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.023205706$ |
$1242.033241$ |
1.789147965 |
\( \frac{86056}{9} a^{3} + 10640 a^{2} - \frac{484492}{9} a - \frac{644893}{9} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 2 a^{2} + 3 a - 6\) , \( a^{2} - 3\) , \( a^{2} - 3 a\) , \( -3 a^{3} + 5 a^{2} + 13 a - 18\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-6\right){x}^{2}+\left(a^{2}-3a\right){x}-3a^{3}+5a^{2}+13a-18$ |
| 10.1-a1 |
10.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{2} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.133654013$ |
$934.3458629$ |
1.937973907 |
\( \frac{1871833}{200} a^{3} - \frac{2780429}{200} a^{2} - \frac{4481327}{100} a + \frac{11733669}{200} \) |
\( \bigl[a^{2} - 4\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a^{3} - 3 a\) , \( a^{3} - 2 a^{2} - 6 a + 7\) , \( -2 a^{2} - 2 a + 4\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(a^{3}-2a^{2}-6a+7\right){x}-2a^{2}-2a+4$ |
| 10.1-a2 |
10.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{2} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.133654013$ |
$934.3458629$ |
1.937973907 |
\( -\frac{4782096383833}{200} a^{3} - \frac{3419314585571}{200} a^{2} + \frac{12339340902127}{100} a + \frac{28483439974731}{200} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - a - 3\) , \( a^{3} - 4 a - 1\) , \( 571 a^{3} + 673 a^{2} - 3201 a - 4346\) , \( -17894 a^{3} - 21137 a^{2} + 100288 a + 136346\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(571a^{3}+673a^{2}-3201a-4346\right){x}-17894a^{3}-21137a^{2}+100288a+136346$ |
| 10.1-a3 |
10.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{6} \cdot 5^{4} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.066827006$ |
$1868.691725$ |
1.937973907 |
\( -\frac{2455822223}{40000} a^{3} - \frac{491069701}{40000} a^{2} + \frac{5731561737}{20000} a + \frac{10191963861}{40000} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - a - 3\) , \( a^{3} - 4 a - 1\) , \( 36 a^{3} + 43 a^{2} - 201 a - 271\) , \( -303 a^{3} - 357 a^{2} + 1700 a + 2309\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(36a^{3}+43a^{2}-201a-271\right){x}-303a^{3}-357a^{2}+1700a+2309$ |
| 10.1-a4 |
10.1-a |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{12} \cdot 5^{8} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.033413503$ |
$467.1729314$ |
1.937973907 |
\( \frac{19886782447697}{1600000000} a^{3} + \frac{48238358525339}{1600000000} a^{2} - \frac{11581007086743}{800000000} a - \frac{73001261517579}{1600000000} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} - 4 a\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 18 a^{3} - 23 a^{2} - 75 a + 94\) , \( 45 a^{3} - 57 a^{2} - 208 a + 250\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(18a^{3}-23a^{2}-75a+94\right){x}+45a^{3}-57a^{2}-208a+250$ |
| 10.1-b1 |
10.1-b |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{8} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$695.2751154$ |
0.599435464 |
\( \frac{3677999808337}{1562500} a^{3} + \frac{9321449560019}{1562500} a^{2} - \frac{1998201005153}{781250} a - \frac{14348708973559}{1562500} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} + a - 5\) , \( a^{3} - 3 a\) , \( 6 a^{3} + 5 a^{2} - 28 a - 30\) , \( -18 a^{3} - 14 a^{2} + 103 a + 118\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(6a^{3}+5a^{2}-28a-30\right){x}-18a^{3}-14a^{2}+103a+118$ |
| 10.1-b2 |
10.1-b |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{6} \cdot 5^{24} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$8.583643401$ |
0.599435464 |
\( -\frac{29005755874006898720221503}{3814697265625000000} a^{3} + \frac{42696248800874354452784939}{3814697265625000000} a^{2} + \frac{70095813011228622435928057}{1907348632812500000} a - \frac{177344621399979778445513179}{3814697265625000000} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a\) , \( 0\) , \( 121 a^{3} - 269 a^{2} - 280 a + 557\) , \( 1618 a^{3} - 3459 a^{2} - 3958 a + 6828\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}={x}^{3}+a{x}^{2}+\left(121a^{3}-269a^{2}-280a+557\right){x}+1618a^{3}-3459a^{2}-3958a+6828$ |
| 10.1-b3 |
10.1-b |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{12} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$17.16728680$ |
0.599435464 |
\( \frac{773459683099171878882073}{1953125000} a^{3} - \frac{1138472237556322588648349}{1953125000} a^{2} - \frac{1869237787797014069371387}{976562500} a + \frac{4729284346873430897617189}{1953125000} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 2 a^{2} + 5 a - 8\) , \( a\) , \( -129 a^{3} + 200 a^{2} + 631 a - 820\) , \( -1291 a^{3} + 1916 a^{2} + 6252 a - 7941\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-8\right){x}^{2}+\left(-129a^{3}+200a^{2}+631a-820\right){x}-1291a^{3}+1916a^{2}+6252a-7941$ |
| 10.1-b4 |
10.1-b |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{4} \) |
$15.35714$ |
$(-a^2-a+2), (a^2-4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1390.550230$ |
0.599435464 |
\( -\frac{2114034047}{1250} a^{3} - \frac{2565306539}{1250} a^{2} + \frac{5933678643}{625} a + \frac{16459318279}{1250} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 2 a^{2} + 5 a - 8\) , \( a\) , \( a^{3} - 4 a\) , \( a + 1\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-8\right){x}^{2}+\left(a^{3}-4a\right){x}+a+1$ |
| 15.1-a1 |
15.1-a |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{4} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$680.7499800$ |
2.641106408 |
\( \frac{2046650087648}{5625} a^{3} + \frac{552283841089}{625} a^{2} - \frac{2285629342124}{5625} a - \frac{7606351776836}{5625} \) |
\( \bigl[a^{2} - 4\) , \( -a\) , \( a^{3} - 3 a\) , \( 217 a^{3} + 257 a^{2} - 1219 a - 1660\) , \( 3066 a^{3} + 3620 a^{2} - 17186 a - 23362\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}-a{x}^{2}+\left(217a^{3}+257a^{2}-1219a-1660\right){x}+3066a^{3}+3620a^{2}-17186a-23362$ |
| 15.1-a2 |
15.1-a |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5 \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$42.54687375$ |
2.641106408 |
\( -\frac{250045832803687076}{32805} a^{3} - \frac{32814197822446553}{3645} a^{2} + \frac{1401511030950055028}{32805} a + \frac{1905362858183732912}{32805} \) |
\( \bigl[a^{2} - 4\) , \( a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - a^{2} - 3 a + 3\) , \( -2 a^{3} - 2 a^{2} - 2 a + 3\) , \( 109 a^{3} + 248 a^{2} - 129 a - 378\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+3\right){x}^{2}+\left(-2a^{3}-2a^{2}-2a+3\right){x}+109a^{3}+248a^{2}-129a-378$ |
| 15.1-a3 |
15.1-a |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{8} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$170.1874950$ |
2.641106408 |
\( -\frac{822746526929980964}{1171875} a^{3} + \frac{585388891247966744}{390625} a^{2} + \frac{2010657466298368007}{1171875} a - \frac{3469036573051750102}{1171875} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - a - 5\) , \( a + 1\) , \( 56 a^{3} - 126 a^{2} - 136 a + 252\) , \( 568 a^{3} - 1210 a^{2} - 1391 a + 2390\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(56a^{3}-126a^{2}-136a+252\right){x}+568a^{3}-1210a^{2}-1391a+2390$ |
| 15.1-a4 |
15.1-a |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5 \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$680.7499800$ |
2.641106408 |
\( -\frac{179206}{45} a^{3} + \frac{48462}{5} a^{2} + \frac{400603}{45} a - \frac{891668}{45} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 6\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 3 a^{3} + 7 a^{2} - 16 a - 23\) , \( 8 a^{3} + 18 a^{2} - 43 a - 81\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-6\right){x}^{2}+\left(3a^{3}+7a^{2}-16a-23\right){x}+8a^{3}+18a^{2}-43a-81$ |
| 15.1-a5 |
15.1-a |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{2} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$170.1874950$ |
2.641106408 |
\( \frac{219860890320514265756}{75} a^{3} + \frac{177746168617595690624}{25} a^{2} - \frac{245738900753444078303}{75} a - \frac{815862498700743369842}{75} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 1\) , \( 161 a^{3} - 345 a^{2} - 394 a + 686\) , \( 3928 a^{3} - 8383 a^{2} - 9602 a + 16558\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+4\right){x}^{2}+\left(161a^{3}-345a^{2}-394a+686\right){x}+3928a^{3}-8383a^{2}-9602a+16558$ |
| 15.1-a6 |
15.1-a |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{2} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$680.7499800$ |
2.641106408 |
\( -\frac{1681210912}{2025} a^{3} - \frac{219604441}{225} a^{2} + \frac{9459598681}{2025} a + \frac{12851596759}{2025} \) |
\( \bigl[1\) , \( -a^{3} + 2 a^{2} + 3 a - 6\) , \( a^{3} - 4 a\) , \( -5 a^{3} + 8 a^{2} + 20 a - 27\) , \( -3 a^{3} + 5 a^{2} + 11 a - 18\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-6\right){x}^{2}+\left(-5a^{3}+8a^{2}+20a-27\right){x}-3a^{3}+5a^{2}+11a-18$ |
| 15.1-b1 |
15.1-b |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{8} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.114464404$ |
$969.3218754$ |
3.443711140 |
\( \frac{17171358992076182}{31640625} a^{3} + \frac{4627492315307726}{3515625} a^{2} - \frac{19192191196876541}{31640625} a - \frac{63721206350304224}{31640625} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} + a - 5\) , \( a + 1\) , \( 59 a^{3} + 43 a^{2} - 297 a - 336\) , \( -362 a^{3} - 353 a^{2} + 1974 a + 2513\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(59a^{3}+43a^{2}-297a-336\right){x}-362a^{3}-353a^{2}+1974a+2513$ |
| 15.1-b2 |
15.1-b |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{2} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.457857617$ |
$484.6609377$ |
3.443711140 |
\( -\frac{1585877524}{75} a^{3} - \frac{624946021}{25} a^{2} + \frac{8890552387}{75} a + \frac{12090783793}{75} \) |
\( \bigl[a\) , \( a^{3} - 5 a - 1\) , \( a^{2} + a - 4\) , \( 9 a^{3} - 16 a^{2} - 43 a + 67\) , \( -103 a^{3} + 152 a^{2} + 497 a - 634\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(9a^{3}-16a^{2}-43a+67\right){x}-103a^{3}+152a^{2}+497a-634$ |
| 15.1-b3 |
15.1-b |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{4} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.228928808$ |
$242.3304688$ |
3.443711140 |
\( \frac{36632995913538418874182}{5625} a^{3} + \frac{9871961401553127278601}{625} a^{2} - \frac{40944763451377812214291}{5625} a - \frac{135938172265360896370849}{5625} \) |
\( \bigl[1\) , \( a^{3} - 2 a^{2} - 4 a + 7\) , \( a^{3} - 3 a - 1\) , \( -13 a^{3} - 37 a^{2} + 11 a + 68\) , \( 94 a^{3} + 219 a^{2} - 110 a - 322\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+7\right){x}^{2}+\left(-13a^{3}-37a^{2}+11a+68\right){x}+94a^{3}+219a^{2}-110a-322$ |
| 15.1-b4 |
15.1-b |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{4} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.228928808$ |
$969.3218754$ |
3.443711140 |
\( -\frac{2895578035432}{5625} a^{3} + \frac{686752077649}{625} a^{2} + \frac{7076356989916}{5625} a - \frac{12209050517276}{5625} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} + a - 3\) , \( -7 a^{3} - 14 a^{2} + a + 13\) , \( 25 a^{3} + 61 a^{2} - 45 a - 120\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-7a^{3}-14a^{2}+a+13\right){x}+25a^{3}+61a^{2}-45a-120$ |
| 15.1-b5 |
15.1-b |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{2} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$0.457857617$ |
$121.1652344$ |
3.443711140 |
\( -\frac{199094538047358564326}{75} a^{3} + \frac{141656912498753537246}{25} a^{2} + \frac{486554371526762489063}{75} a - \frac{839464189520604318868}{75} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} + a - 3\) , \( -12 a^{3} - 39 a^{2} + 16 a + 73\) , \( -41 a^{3} - 106 a^{2} + a + 99\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-12a^{3}-39a^{2}+16a+73\right){x}-41a^{3}-106a^{2}+a+99$ |
| 15.1-b6 |
15.1-b |
$6$ |
$8$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{16} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.057232202$ |
$242.3304688$ |
3.443711140 |
\( \frac{2062005521815549687682}{1001129150390625} a^{3} - \frac{696113221281477144649}{111236572265625} a^{2} - \frac{9051133442912648113541}{1001129150390625} a + \frac{29278092512979380468401}{1001129150390625} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( a^{3} - 2 a^{2} - 5 a + 7\) , \( a^{2} - 3\) , \( -113 a^{3} + 151 a^{2} + 536 a - 657\) , \( 726 a^{3} - 922 a^{2} - 3902 a + 4698\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+7\right){x}^{2}+\left(-113a^{3}+151a^{2}+536a-657\right){x}+726a^{3}-922a^{2}-3902a+4698$ |
| 15.1-c1 |
15.1-c |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{28} \cdot 5^{5} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \cdot 7 \) |
$1$ |
$12.24268436$ |
3.324856873 |
\( \frac{34459066371485137429502865701356}{71489976421753125} a^{3} + \frac{9286124836366533753288974212258}{7943330713528125} a^{2} - \frac{38514958583176697866664695317028}{71489976421753125} a - \frac{127871127732986264791387575120217}{71489976421753125} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( a\) , \( 0\) , \( 746 a^{3} + 868 a^{2} - 4207 a - 5705\) , \( -25997 a^{3} - 30776 a^{2} + 145396 a + 197778\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}={x}^{3}+a{x}^{2}+\left(746a^{3}+868a^{2}-4207a-5705\right){x}-25997a^{3}-30776a^{2}+145396a+197778$ |
| 15.1-c2 |
15.1-c |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{7} \cdot 5^{20} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \cdot 7 \) |
$1$ |
$12.24268436$ |
3.324856873 |
\( -\frac{257744452774563336093797729144596}{208568572998046875} a^{3} + \frac{61128888402017957923090487557922}{23174285888671875} a^{2} + \frac{629885136309116047734538228605148}{208568572998046875} a - \frac{1086756273025335961926686912829353}{208568572998046875} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( a\) , \( 0\) , \( 576 a^{3} + 548 a^{2} - 3097 a - 3915\) , \( 15575 a^{3} + 19386 a^{2} - 88244 a - 122238\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}={x}^{3}+a{x}^{2}+\left(576a^{3}+548a^{2}-3097a-3915\right){x}+15575a^{3}+19386a^{2}-88244a-122238$ |
| 15.1-c3 |
15.1-c |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{10} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 5 \cdot 7 \) |
$1$ |
$48.97073747$ |
3.324856873 |
\( -\frac{295868428911559113928}{46708681640625} a^{3} + \frac{181015927651769833496}{5189853515625} a^{2} + \frac{1000709717604214730764}{46708681640625} a - \frac{2966996167275463595579}{46708681640625} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( a\) , \( 0\) , \( 61 a^{3} + 68 a^{2} - 332 a - 450\) , \( -163 a^{3} - 177 a^{2} + 896 a + 1170\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}={x}^{3}+a{x}^{2}+\left(61a^{3}+68a^{2}-332a-450\right){x}-163a^{3}-177a^{2}+896a+1170$ |
| 15.1-c4 |
15.1-c |
$4$ |
$4$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{7} \cdot 5^{5} \) |
$16.15555$ |
$(a^2-a-3), (a^2-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \cdot 7 \) |
$1$ |
$48.97073747$ |
3.324856873 |
\( \frac{502592162337840848}{6834375} a^{3} - \frac{82197380355511936}{759375} a^{2} - \frac{2429252047467351224}{6834375} a + \frac{3073077137409412639}{6834375} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( a\) , \( 0\) , \( -14 a^{3} - 12 a^{2} + 83 a + 95\) , \( -29 a^{3} - 28 a^{2} + 166 a + 200\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}={x}^{3}+a{x}^{2}+\left(-14a^{3}-12a^{2}+83a+95\right){x}-29a^{3}-28a^{2}+166a+200$ |
| 16.1-a1 |
16.1-a |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{9} \) |
$16.28641$ |
$(-a^2-a+2), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$450.0014497$ |
1.745871094 |
\( -\frac{309893647}{64} a^{3} + \frac{33184763}{64} a^{2} + \frac{687407801}{32} a + \frac{963413045}{64} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{2} + a + 5\) , \( a^{2} + a - 4\) , \( 45 a^{3} + 47 a^{2} - 249 a - 317\) , \( -272 a^{3} - 323 a^{2} + 1526 a + 2077\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(45a^{3}+47a^{2}-249a-317\right){x}-272a^{3}-323a^{2}+1526a+2077$ |
| 16.1-a2 |
16.1-a |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{9} \) |
$16.28641$ |
$(-a^2-a+2), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$450.0014497$ |
1.745871094 |
\( \frac{2931721}{8} a^{3} - \frac{4330539}{8} a^{2} - \frac{7080185}{4} a + \frac{17987743}{8} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 4 a + 8\) , \( 0\) , \( 7 a^{3} - 14 a^{2} - 22 a + 36\) , \( 16 a^{3} - 33 a^{2} - 43 a + 71\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+8\right){x}^{2}+\left(7a^{3}-14a^{2}-22a+36\right){x}+16a^{3}-33a^{2}-43a+71$ |
| 16.1-a3 |
16.1-a |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{11} \) |
$16.28641$ |
$(-a^2-a+2), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$450.0014497$ |
1.745871094 |
\( \frac{511731662441016533}{8} a^{3} + \frac{1241125816830348625}{8} a^{2} - \frac{571963372001580127}{8} a - \frac{1898940153481625825}{8} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( -a^{3} + 3 a + 1\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 24 a^{3} - 62 a^{2} - 61 a + 122\) , \( 2 a^{3} + 12 a^{2} - a - 18\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(24a^{3}-62a^{2}-61a+122\right){x}+2a^{3}+12a^{2}-a-18$ |
| 16.1-a4 |
16.1-a |
$4$ |
$6$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{19} \) |
$16.28641$ |
$(-a^2-a+2), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$450.0014497$ |
1.745871094 |
\( -\frac{3447353197}{64} a^{3} - \frac{4177386519}{32} a^{2} + \frac{1937942843}{32} a + \frac{12812714531}{64} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 4\) , \( a^{3} - 2 a^{2} - 4 a + 7\) , \( a\) , \( 213 a^{3} + 238 a^{2} - 1205 a - 1588\) , \( 4044 a^{3} + 4776 a^{2} - 22642 a - 30762\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+4\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+7\right){x}^{2}+\left(213a^{3}+238a^{2}-1205a-1588\right){x}+4044a^{3}+4776a^{2}-22642a-30762$ |
| 16.1-b1 |
16.1-b |
$2$ |
$2$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{13} \) |
$16.28641$ |
$(-a^2-a+2), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.808333376$ |
$182.9696300$ |
2.295238257 |
\( -\frac{22009440895}{1024} a^{3} + \frac{47809965227}{1024} a^{2} + \frac{25400978969}{512} a - \frac{90433804059}{1024} \) |
\( \bigl[a\) , \( a^{2} - 3\) , \( 1\) , \( -23 a^{3} + 32 a^{2} + 122 a - 151\) , \( -127 a^{3} + 177 a^{2} + 652 a - 808\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-23a^{3}+32a^{2}+122a-151\right){x}-127a^{3}+177a^{2}+652a-808$ |
| 16.1-b2 |
16.1-b |
$2$ |
$2$ |
4.4.16609.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{11} \) |
$16.28641$ |
$(-a^2-a+2), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.404166688$ |
$365.9392600$ |
2.295238257 |
\( -\frac{4775263}{32} a^{3} - \frac{5837741}{32} a^{2} + \frac{13462297}{16} a + \frac{37168965}{32} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} + a - 3\) , \( -5 a^{3} + 9 a^{2} + 25 a - 32\) , \( -17 a^{3} + 14 a^{2} + 72 a - 77\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-5a^{3}+9a^{2}+25a-32\right){x}-17a^{3}+14a^{2}+72a-77$ |
*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.
