| 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 |
| 4.1-a1 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{9} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.856387478$ |
$528.7585263$ |
2.439212162 |
\( \frac{55551499}{64} a^{3} + \frac{5153177}{16} a^{2} - \frac{31243571}{16} a - \frac{837273}{64} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 4 a - 3\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 14 a^{2} - 19 a - 9\) , \( 2 a^{3} + 8 a^{2} - 13 a - 5\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-4a-3\right){x}^{2}+\left(14a^{2}-19a-9\right){x}+2a^{3}+8a^{2}-13a-5$ |
| 4.1-a2 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{11} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$2.569162434$ |
$6.527883041$ |
2.439212162 |
\( \frac{78090946968383133271389}{512} a^{3} + \frac{54158039788414000050149}{512} a^{2} - \frac{166487742800179523590269}{512} a - \frac{28992098126199977770199}{256} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 4 a - 3\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 40 a^{3} - 171 a^{2} + 111 a + 51\) , \( 485 a^{3} - 2100 a^{2} + 1402 a + 705\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-4a-3\right){x}^{2}+\left(40a^{3}-171a^{2}+111a+51\right){x}+485a^{3}-2100a^{2}+1402a+705$ |
| 4.1-a3 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.712774956$ |
$264.3792631$ |
2.439212162 |
\( -\frac{137939962377536181}{16777216} a^{3} + \frac{509484747527592835}{16777216} a^{2} - \frac{311065423629019051}{16777216} a - \frac{81451387571906785}{8388608} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a - 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 5\) , \( 4 a^{3} + 21 a^{2} - 4 a - 41\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a^{3}-5\right){x}+4a^{3}+21a^{2}-4a-41$ |
| 4.1-a4 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{22} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$1.284581217$ |
$13.05576608$ |
2.439212162 |
\( \frac{761611006401787}{262144} a^{3} + \frac{528825605871283}{262144} a^{2} - \frac{1631943548046683}{262144} a - \frac{277572643256305}{131072} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a^{3} - 5 a - 2\) , \( -339 a^{3} + 815 a^{2} + 1078 a - 2185\) , \( -7153 a^{3} + 16767 a^{2} + 23042 a - 43919\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-339a^{3}+815a^{2}+1078a-2185\right){x}-7153a^{3}+16767a^{2}+23042a-43919$ |
| 4.1-a5 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.428193739$ |
$1057.517052$ |
2.439212162 |
\( -\frac{775099}{4096} a^{3} + \frac{325913}{1024} a^{2} + \frac{1405713}{1024} a + \frac{9152113}{4096} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a^{3} - 5 a - 2\) , \( -4 a^{3} + 15 a^{2} + 8 a - 45\) , \( -2 a^{3} + 9 a^{2} + 4 a - 27\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-4a^{3}+15a^{2}+8a-45\right){x}-2a^{3}+9a^{2}+4a-27$ |
| 4.1-a6 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{17} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$2.569162434$ |
$1.631970760$ |
2.439212162 |
\( -\frac{280401766250925363}{512} a^{3} + \frac{1034629287445097955}{512} a^{2} - \frac{629865483106281211}{512} a - \frac{20638781304796925}{32} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -213 a^{3} + 305 a^{2} + 615 a - 867\) , \( -2875 a^{3} + 4341 a^{2} + 8409 a - 12175\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-213a^{3}+305a^{2}+615a-867\right){x}-2875a^{3}+4341a^{2}+8409a-12175$ |
| 4.1-a7 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{27} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.856387478$ |
$132.1896315$ |
2.439212162 |
\( -\frac{237926148587}{16777216} a^{3} + \frac{189399317433}{4194304} a^{2} - \frac{64016324079}{4194304} a - \frac{147860638399}{16777216} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 12 a^{3} - 25 a^{2} - 35 a + 73\) , \( 2 a^{3} - 31 a^{2} - 13 a + 77\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(12a^{3}-25a^{2}-35a+73\right){x}+2a^{3}-31a^{2}-13a+77$ |
| 4.1-a8 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{19} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$5.138324868$ |
$3.263941520$ |
2.439212162 |
\( \frac{242040755788705504414306661066662661}{262144} a^{3} - \frac{563888926547782432185802163326932659}{262144} a^{2} - \frac{782233519623267834773636928774173349}{262144} a + \frac{734063714311722039417478948314159089}{131072} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{3} - 5 a - 1\) , \( -187 a^{3} + 626 a^{2} - 349 a - 222\) , \( 12566 a^{3} - 46564 a^{2} + 28613 a + 14737\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(-187a^{3}+626a^{2}-349a-222\right){x}+12566a^{3}-46564a^{2}+28613a+14737$ |
| 4.1-a9 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.712774956$ |
$264.3792631$ |
2.439212162 |
\( -\frac{1211116029998867}{64} a^{3} + \frac{811492257554189}{64} a^{2} + \frac{5924196660024347}{64} a + \frac{911455142569285}{32} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{3} - 5 a - 1\) , \( 13 a^{3} - 84 a^{2} + 46 a + 38\) , \( -471 a^{3} + 1907 a^{2} - 1237 a - 623\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(13a^{3}-84a^{2}+46a+38\right){x}-471a^{3}+1907a^{2}-1237a-623$ |
| 4.1-a10 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.856387478$ |
$1057.517052$ |
2.439212162 |
\( -\frac{6814982421}{4096} a^{3} + \frac{4701382115}{4096} a^{2} + \frac{32854186741}{4096} a + \frac{5318720799}{2048} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{3} - 5 a - 1\) , \( 33 a^{3} - 119 a^{2} + 61 a + 48\) , \( -301 a^{3} + 1114 a^{2} - 690 a - 349\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(33a^{3}-119a^{2}+61a+48\right){x}-301a^{3}+1114a^{2}-690a-349$ |
| 4.1-a11 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{38} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$2.569162434$ |
$13.05576608$ |
2.439212162 |
\( \frac{25244062543252502649437931}{68719476736} a^{3} - \frac{58811778546102253618393565}{68719476736} a^{2} - \frac{81584408494612303988948299}{68719476736} a + \frac{76560454679459868202267199}{34359738368} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{3} - 5 a - 1\) , \( 138 a^{3} - 549 a^{2} + 341 a + 188\) , \( 2457 a^{3} - 9401 a^{2} + 5861 a + 3045\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(138a^{3}-549a^{2}+341a+188\right){x}+2457a^{3}-9401a^{2}+5861a+3045$ |
| 4.1-a12 |
4.1-a |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{73} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$5.138324868$ |
$3.263941520$ |
2.439212162 |
\( -\frac{34604215282042281704676226101}{4722366482869645213696} a^{3} + \frac{80619066741497191268959792643}{4722366482869645213696} a^{2} + \frac{111827495038782804569975667157}{4722366482869645213696} a - \frac{104940827717958463993588143201}{2361183241434822606848} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -81 a^{3} + 24 a^{2} + 464 a + 146\) , \( 642 a^{3} - 317 a^{2} - 3434 a - 1078\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-81a^{3}+24a^{2}+464a+146\right){x}+642a^{3}-317a^{2}-3434a-1078$ |
| 4.1-b1 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{17} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$2.569162434$ |
$1.631970760$ |
2.439212162 |
\( \frac{280401766250925363}{512} a^{3} + \frac{96711994346160933}{256} a^{2} - \frac{299093896515569305}{256} a - \frac{205858462788859419}{512} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 4\) , \( a^{2} - a - 3\) , \( 210 a^{3} - 330 a^{2} - 576 a - 161\) , \( 3369 a^{3} - 5116 a^{2} - 9720 a - 2540\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-4\right){x}^{2}+\left(210a^{3}-330a^{2}-576a-161\right){x}+3369a^{3}-5116a^{2}-9720a-2540$ |
| 4.1-b2 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{27} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.856387478$ |
$132.1896315$ |
2.439212162 |
\( \frac{237926148587}{16777216} a^{3} + \frac{43818823971}{16777216} a^{2} - \frac{545350797387}{16777216} a + \frac{57872593215}{8388608} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 4\) , \( a^{2} - a - 3\) , \( -15 a^{3} + 15 a^{2} + 59 a + 24\) , \( -38 a^{3} + 18 a^{2} + 205 a + 60\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-4\right){x}^{2}+\left(-15a^{3}+15a^{2}+59a+24\right){x}-38a^{3}+18a^{2}+205a+60$ |
| 4.1-b3 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{19} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$5.138324868$ |
$3.263941520$ |
2.439212162 |
\( -\frac{242040755788705504414306661066662661}{262144} a^{3} + \frac{40558335204583520264279454968263831}{65536} a^{2} + \frac{295972276338179046475580318057012671}{65536} a + \frac{364045738241099316289825465593874831}{262144} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 4 a - 1\) , \( 186 a^{3} + 62 a^{2} - 334 a - 122\) , \( -12129 a^{3} - 8459 a^{2} + 25756 a + 8984\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(186a^{3}+62a^{2}-334a-122\right){x}-12129a^{3}-8459a^{2}+25756a+8984$ |
| 4.1-b4 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.712774956$ |
$264.3792631$ |
2.439212162 |
\( \frac{1211116029998867}{64} a^{3} - \frac{705463958110603}{16} a^{2} - \frac{978458271284031}{16} a + \frac{7347483172718239}{64} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 4 a - 1\) , \( -14 a^{3} - 48 a^{2} + 91 a + 23\) , \( 398 a^{3} + 526 a^{2} - 1080 a - 392\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-14a^{3}-48a^{2}+91a+23\right){x}+398a^{3}+526a^{2}-1080a-392$ |
| 4.1-b5 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.856387478$ |
$1057.517052$ |
2.439212162 |
\( \frac{6814982421}{4096} a^{3} - \frac{3935891287}{1024} a^{2} - \frac{5453000927}{1024} a + \frac{41378028033}{4096} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 4 a - 1\) , \( -34 a^{3} - 23 a^{2} + 86 a + 33\) , \( 213 a^{3} + 158 a^{2} - 441 a - 154\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-34a^{3}-23a^{2}+86a+33\right){x}+213a^{3}+158a^{2}-441a-154$ |
| 4.1-b6 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{38} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$2.569162434$ |
$13.05576608$ |
2.439212162 |
\( -\frac{25244062543252502649437931}{68719476736} a^{3} + \frac{4230102270913813582480057}{17179869184} a^{2} + \frac{30868944489264825819355409}{17179869184} a + \frac{37968784861457681446630465}{68719476736} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 4 a - 1\) , \( -139 a^{3} - 138 a^{2} + 351 a + 128\) , \( -2870 a^{3} - 2238 a^{2} + 6384 a + 2244\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-139a^{3}-138a^{2}+351a+128\right){x}-2870a^{3}-2238a^{2}+6384a+2244$ |
| 4.1-b7 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{22} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$1.284581217$ |
$13.05576608$ |
2.439212162 |
\( -\frac{761611006401787}{262144} a^{3} + \frac{703414656269161}{65536} a^{2} - \frac{427635170725311}{65536} a - \frac{896652222286223}{262144} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{2} - 2 a - 2\) , \( 1\) , \( 341 a^{3} - 207 a^{2} - 1698 a - 620\) , \( 6813 a^{3} - 4488 a^{2} - 33422 a - 10636\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(341a^{3}-207a^{2}-1698a-620\right){x}+6813a^{3}-4488a^{2}-33422a-10636$ |
| 4.1-b8 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.428193739$ |
$1057.517052$ |
2.439212162 |
\( \frac{775099}{4096} a^{3} - \frac{1021645}{4096} a^{2} - \frac{5904859}{4096} a + \frac{7651759}{2048} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{2} - 2 a - 2\) , \( 1\) , \( 6 a^{3} - 2 a^{2} - 33 a - 15\) , \( -3 a^{3} + 2 a^{2} + 14 a + 6\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(6a^{3}-2a^{2}-33a-15\right){x}-3a^{3}+2a^{2}+14a+6$ |
| 4.1-b9 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.712774956$ |
$264.3792631$ |
2.439212162 |
\( \frac{137939962377536181}{16777216} a^{3} + \frac{23916215098746073}{4194304} a^{2} - \frac{73521046073389519}{4194304} a - \frac{102423413622775967}{16777216} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 14 a + 2\) , \( -3 a^{3} + 31 a^{2} - 55 a - 18\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(a^{3}-14a+2\right){x}-3a^{3}+31a^{2}-55a-18$ |
| 4.1-b10 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{9} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.856387478$ |
$528.7585263$ |
2.439212162 |
\( -\frac{55551499}{64} a^{3} + \frac{187267205}{64} a^{2} - \frac{82905629}{64} a - \frac{24823675}{32} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 4 a - 1\) , \( 3 a^{3} - 4 a + 6\) , \( a^{3} + 4\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(3a^{3}-4a+6\right){x}+a^{3}+4$ |
| 4.1-b11 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{11} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$2.569162434$ |
$6.527883041$ |
2.439212162 |
\( -\frac{78090946968383133271389}{512} a^{3} + \frac{72107720173390849966079}{128} a^{2} - \frac{44025294420449469081049}{128} a - \frac{92222952295782345809129}{512} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 4 a - 1\) , \( -37 a^{3} - 65 a^{2} + 116 a + 51\) , \( -522 a^{3} - 534 a^{2} + 1277 a + 464\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-37a^{3}-65a^{2}+116a+51\right){x}-522a^{3}-534a^{2}+1277a+464$ |
| 4.1-b12 |
4.1-b |
$12$ |
$24$ |
4.4.15317.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{73} \) |
$13.15173$ |
$(-a), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$5.138324868$ |
$3.263941520$ |
2.439212162 |
\( \frac{34604215282042281704676226101}{4722366482869645213696} a^{3} - \frac{5798394776157413461267221415}{1180591620717411303424} a^{2} - \frac{42313245668912585498466643535}{1180591620717411303424} a - \frac{52039308937679213852917052703}{4722366482869645213696} \) |
\( \bigl[1\) , \( -a^{3} + 2 a^{2} + 4 a - 5\) , \( a^{3} - 5 a - 2\) , \( 80 a^{3} - 218 a^{2} - 265 a + 553\) , \( -641 a^{3} + 1608 a^{2} + 2136 a - 4187\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-5\right){x}^{2}+\left(80a^{3}-218a^{2}-265a+553\right){x}-641a^{3}+1608a^{2}+2136a-4187$ |
*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.
