| 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 |
| 22.1-a1 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{4} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1596.650924$ |
0.841626760 |
\( -\frac{201814041699}{1936} a^{3} + \frac{35160513793}{1936} a^{2} + \frac{418144402265}{968} a + \frac{488783259193}{1936} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} - 2 a^{2} + 4 a + 4\) , \( -a^{3} + 4 a\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{3}-2a^{2}+4a+4\right){x}-a^{3}+4a$ |
| 22.1-a2 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{2} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$199.5813655$ |
0.841626760 |
\( -\frac{2581686241508983459}{44} a^{3} + \frac{449769625560286565}{44} a^{2} + \frac{5349078887234667531}{22} a + \frac{6252687182257486625}{44} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 4 a^{3} - 22 a^{2} - a + 14\) , \( 6 a^{3} - 65 a^{2} + 6 a + 36\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(4a^{3}-22a^{2}-a+14\right){x}+6a^{3}-65a^{2}+6a+36$ |
| 22.1-a3 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{16} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$399.1627311$ |
0.841626760 |
\( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( -20 a^{3} + 35 a^{2} + 56 a - 64\) , \( 299 a^{3} - 501 a^{2} - 856 a + 876\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(-20a^{3}+35a^{2}+56a-64\right){x}+299a^{3}-501a^{2}-856a+876$ |
| 22.1-a4 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{48} \cdot 11^{6} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$4.927934952$ |
0.841626760 |
\( -\frac{1502268147076192175039019336779}{498650091216506454016} a^{3} + \frac{261719896030458847171340772313}{498650091216506454016} a^{2} + \frac{3112599543413066339294706687465}{249325045608253227008} a + \frac{3638403917729845697875388025793}{498650091216506454016} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( 255 a^{3} - 345 a^{2} - 799 a + 451\) , \( -6627 a^{3} + 11701 a^{2} + 18463 a - 21549\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(255a^{3}-345a^{2}-799a+451\right){x}-6627a^{3}+11701a^{2}+18463a-21549$ |
| 22.1-a5 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{6} \cdot 11^{3} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$9.855869904$ |
0.841626760 |
\( \frac{174550817373650134723}{85184} a^{3} - \frac{293183090937363721025}{85184} a^{2} - \frac{249471689806794754865}{42592} a + \frac{513654285309827502679}{85184} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 1\) , \( -13 a^{3} + 17 a^{2} + 48 a - 41\) , \( -75 a^{3} + 116 a^{2} + 245 a - 240\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(-13a^{3}+17a^{2}+48a-41\right){x}-75a^{3}+116a^{2}+245a-240$ |
| 22.1-a6 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{2} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$798.3254622$ |
0.841626760 |
\( \frac{2490151}{44} a^{3} - \frac{3891025}{44} a^{2} - \frac{3683329}{22} a + \frac{6322871}{44} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 1\) , \( 2 a^{3} - 3 a^{2} - 7 a + 4\) , \( a^{3} - a^{2} - 4 a\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(2a^{3}-3a^{2}-7a+4\right){x}+a^{3}-a^{2}-4a$ |
| 22.1-a7 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{4} \cdot 11^{8} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$399.1627311$ |
0.841626760 |
\( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -29 a^{3} - 37 a^{2} + 29 a + 24\) , \( 247 a^{3} + 345 a^{2} - 177 a - 202\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(-29a^{3}-37a^{2}+29a+24\right){x}+247a^{3}+345a^{2}-177a-202$ |
| 22.1-a8 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{12} \cdot 11^{6} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$19.71173980$ |
0.841626760 |
\( -\frac{20157112611959390547}{7256313856} a^{3} + \frac{50708561159662371377}{7256313856} a^{2} + \frac{1953876284050968001}{3628156928} a - \frac{26455281535659018279}{7256313856} \) |
\( \bigl[a^{2} - 1\) , \( 0\) , \( a^{3} - 4 a - 1\) , \( -65 a^{3} + 111 a^{2} + 209 a - 254\) , \( -552 a^{3} + 882 a^{2} + 1709 a - 1680\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-65a^{3}+111a^{2}+209a-254\right){x}-552a^{3}+882a^{2}+1709a-1680$ |
| 22.1-a9 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{6} \cdot 11^{3} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$36$ |
\( 2 \) |
$1$ |
$2.463967476$ |
0.841626760 |
\( -\frac{5006065775663608963358467}{85184} a^{3} + \frac{12557622075673248231038785}{85184} a^{2} + \frac{540663280751702976925425}{42592} a - \frac{6637226716732893866394967}{85184} \) |
\( \bigl[a^{2} - 1\) , \( 0\) , \( a^{3} - 4 a - 1\) , \( 210 a^{3} - 374 a^{2} - 201 a - 214\) , \( 1496 a^{3} - 6224 a^{2} + 4609 a + 4096\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(210a^{3}-374a^{2}-201a-214\right){x}+1496a^{3}-6224a^{2}+4609a+4096$ |
| 22.1-a10 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{12} \cdot 11^{24} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$4.927934952$ |
0.841626760 |
\( \frac{60659117288902836130412254108283}{40344505040107975043939700736} a^{3} - \frac{86300110386401779908460150637033}{40344505040107975043939700736} a^{2} - \frac{97109108201994369690495482301689}{20172252520053987521969850368} a + \frac{184521640869151181066006363788207}{40344505040107975043939700736} \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -115 a^{3} + 187 a^{2} + 362 a - 350\) , \( 926 a^{3} - 1541 a^{2} - 2691 a + 2732\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-115a^{3}+187a^{2}+362a-350\right){x}+926a^{3}-1541a^{2}-2691a+2732$ |
| 22.1-a11 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{24} \cdot 11^{12} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$19.71173980$ |
0.841626760 |
\( -\frac{2125946353940568480801651}{52654090776777588736} a^{3} + \frac{2559556155064329492768145}{52654090776777588736} a^{2} + \frac{2498155651166365388592929}{26327045388388794368} a + \frac{2080764888610005913150009}{52654090776777588736} \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -30 a^{3} + 42 a^{2} + 102 a - 90\) , \( -111 a^{3} + 178 a^{2} + 344 a - 362\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-30a^{3}+42a^{2}+102a-90\right){x}-111a^{3}+178a^{2}+344a-362$ |
| 22.1-a12 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{8} \cdot 11^{4} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1596.650924$ |
0.841626760 |
\( \frac{1878243643917}{3748096} a^{3} + \frac{1364709508625}{3748096} a^{2} - \frac{1139759245535}{1874048} a - \frac{14164000839}{3748096} \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -10 a^{3} + 12 a^{2} + 32 a - 15\) , \( 16 a^{3} - 29 a^{2} - 45 a + 56\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-10a^{3}+12a^{2}+32a-15\right){x}+16a^{3}-29a^{2}-45a+56$ |
| 22.1-b1 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2 \cdot 11^{20} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.679157918$ |
1.610989990 |
\( \frac{1573390889429092537195325472495367149}{1345499989865120018402} a^{3} - \frac{2642735286096795180305969741051658561}{1345499989865120018402} a^{2} - \frac{2248723264231462392984836360638223253}{672749994932560009201} a + \frac{4630049775522972966851131993588823229}{1345499989865120018402} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 574 a^{3} - 1057 a^{2} - 549 a - 385\) , \( 14427 a^{3} - 29786 a^{2} - 11937 a + 4219\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(574a^{3}-1057a^{2}-549a-385\right){x}+14427a^{3}-29786a^{2}-11937a+4219$ |
| 22.1-b2 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{5} \cdot 11^{4} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$424.4736992$ |
1.610989990 |
\( \frac{19492220557608647265}{468512} a^{3} + \frac{26555029218814954293}{468512} a^{2} - \frac{7618439534802966603}{234256} a - \frac{16502468789812849155}{468512} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 9 a^{3} + 8 a^{2} - 59 a - 35\) , \( -37 a^{3} + 17 a^{2} + 136 a + 71\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(9a^{3}+8a^{2}-59a-35\right){x}-37a^{3}+17a^{2}+136a+71$ |
| 22.1-b3 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{10} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2Cs, 5B.1.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$1697.894796$ |
1.610989990 |
\( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 4 a^{3} - 7 a^{2} - 9 a + 5\) , \( -a^{3} + 2 a^{2} + a - 3\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(4a^{3}-7a^{2}-9a+5\right){x}-a^{3}+2a^{2}+a-3$ |
| 22.1-b4 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 5 \) |
$1$ |
$2.716631675$ |
1.610989990 |
\( -\frac{776940933840351053858986959}{322102} a^{3} + \frac{135355112947607678458237413}{322102} a^{2} + \frac{1609768948326117754076482158}{161051} a + \frac{1881703725424533772239476133}{322102} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -25 a^{3} + 26 a^{2} + 84 a - 75\) , \( -255 a^{3} - 134 a^{2} + 413 a - 111\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(-25a^{3}+26a^{2}+84a-75\right){x}-255a^{3}-134a^{2}+413a-111$ |
| 22.1-b5 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$1697.894796$ |
1.610989990 |
\( -\frac{38015163}{352} a^{3} + \frac{90874953}{352} a^{2} + \frac{7617321}{176} a - \frac{43556751}{352} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 2 a - 3\) , \( a^{3} - 3 a - 1\) , \( -2 a - 2\) , \( 28 a^{3} - 47 a^{2} - 81 a + 81\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-2a-2\right){x}+28a^{3}-47a^{2}-81a+81$ |
| 22.1-b6 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{2} \cdot 11^{10} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2Cs, 5B.1.2 |
$25$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.716631675$ |
1.610989990 |
\( -\frac{64272348420343413978591}{103749698404} a^{3} + \frac{156108841633210925549601}{103749698404} a^{2} + \frac{11245678214829850896543}{51874849202} a - \frac{76993471097806369350591}{103749698404} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 2 a - 3\) , \( a^{2} - a - 1\) , \( -833 a^{3} - 1088 a^{2} + 652 a + 590\) , \( -39688 a^{3} - 53892 a^{2} + 31054 a + 33219\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-833a^{3}-1088a^{2}+652a+590\right){x}-39688a^{3}-53892a^{2}+31054a+33219$ |
| 22.1-b7 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{20} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$424.4736992$ |
1.610989990 |
\( \frac{10003437681517215}{11534336} a^{3} - \frac{16802049412165365}{11534336} a^{2} - \frac{14297353230002373}{5767168} a + \frac{29437605898282179}{11534336} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 4 a - 2\) , \( 0\) , \( 4 a^{3} - 15 a^{2} + 7 a + 9\) , \( 26 a^{3} - 63 a^{2} - 10 a + 33\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(4a^{3}-15a^{2}+7a+9\right){x}+26a^{3}-63a^{2}-10a+33$ |
| 22.1-b8 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{4} \cdot 11^{5} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.679157918$ |
1.610989990 |
\( -\frac{6909632377058601104250692620017}{2576816} a^{3} + \frac{17332682911101563069387220726123}{2576816} a^{2} + \frac{746251602738792012142363943163}{1288408} a - \frac{9161045402238428490227106090477}{2576816} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a\) , \( -506 a^{3} - 392 a^{2} + 539 a - 67\) , \( -15172 a^{3} - 18204 a^{2} + 13108 a + 9055\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-506a^{3}-392a^{2}+539a-67\right){x}-15172a^{3}-18204a^{2}+13108a+9055$ |
| 22.1-c1 |
22.1-c |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{4} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.027949886$ |
$2051.322385$ |
1.087992806 |
\( \frac{52878458037}{1936} a^{3} - \frac{27483917095}{1936} a^{2} - \frac{55514275415}{968} a + \frac{80654280017}{1936} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( -5 a^{3} + 4 a^{2} + 18 a - 5\) , \( -4 a + 10\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-5a^{3}+4a^{2}+18a-5\right){x}-4a+10$ |
| 22.1-c2 |
22.1-c |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{2} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.055899772$ |
$512.8305963$ |
1.087992806 |
\( \frac{85048962626261637}{44} a^{3} + \frac{115966996128885001}{44} a^{2} - \frac{33248386401040127}{22} a - \frac{72060916678769347}{44} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 5 a^{3} - 46 a^{2} + 23 a + 25\) , \( -76 a^{3} + 260 a^{2} + 9 a - 124\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(5a^{3}-46a^{2}+23a+25\right){x}-76a^{3}+260a^{2}+9a-124$ |
| 22.1-c3 |
22.1-c |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{8} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.055899772$ |
$512.8305963$ |
1.087992806 |
\( \frac{3183389426137826703}{2816} a^{3} - \frac{5346958357237086533}{2816} a^{2} - \frac{4549766946734504469}{1408} a + \frac{9367825631231892211}{2816} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( 63 a^{3} - 9 a^{2} - 267 a - 166\) , \( 458 a^{3} - 86 a^{2} - 1890 a - 1084\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(63a^{3}-9a^{2}-267a-166\right){x}+458a^{3}-86a^{2}-1890a-1084$ |
| 22.1-c4 |
22.1-c |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2 \cdot 11^{8} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.027949886$ |
$256.4152981$ |
1.087992806 |
\( -\frac{981714842139174225}{428717762} a^{3} + \frac{171035604423335643}{428717762} a^{2} + \frac{2034041293141012467}{214358881} a + \frac{2377646930850683857}{428717762} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - 2 a - 1\) , \( 0\) , \( 4 a^{3} - 4 a^{2} - 4 a\) , \( 15 a^{3} - 18 a^{2} - a + 12\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(a^{2}-2a-1\right){x}^{2}+\left(4a^{3}-4a^{2}-4a\right){x}+15a^{3}-18a^{2}-a+12$ |
| 22.1-c5 |
22.1-c |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{2} \cdot 11^{4} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.013974943$ |
$2051.322385$ |
1.087992806 |
\( -\frac{1048982343}{58564} a^{3} + \frac{91465773}{58564} a^{2} + \frac{2303862237}{29282} a + \frac{3279047225}{58564} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - 2 a - 1\) , \( 0\) , \( -a^{3} + a^{2} + a\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(a^{2}-2a-1\right){x}^{2}+\left(-a^{3}+a^{2}+a\right){x}$ |
| 22.1-c6 |
22.1-c |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2 \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.027949886$ |
$1025.661192$ |
1.087992806 |
\( \frac{36105}{242} a^{3} + \frac{168125}{242} a^{2} - \frac{39059}{121} a - \frac{72489}{242} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 3 a - 1\) , \( 3 a^{3} - 5 a^{2} - 10 a + 8\) , \( 2 a^{3} - 4 a^{2} - 7 a + 6\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(3a^{3}-5a^{2}-10a+8\right){x}+2a^{3}-4a^{2}-7a+6$ |
*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.
