| 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 |
| 15.1-a1 |
15.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5 \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$194.3145676$ |
2.897556047 |
\( -\frac{17638}{45} a^{3} - \frac{5397}{5} a^{2} - \frac{6598}{9} a - \frac{4736}{45} \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 5 a\) , \( 1\) , \( -2 a\) , \( -a^{2} - a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-2a^{2}-5a\right){x}^{2}-2a{x}-a^{2}-a$ |
| 15.1-b1 |
15.1-b |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{9} \cdot 5^{3} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 1 \) |
$1$ |
$832.0096251$ |
1.550832337 |
\( -\frac{5160440524}{2460375} a^{3} + \frac{1069321939}{273375} a^{2} + \frac{8607014084}{492075} a + \frac{20961784912}{2460375} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 12 a - 1\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( 2 a^{3} - 3 a^{2} - 11 a\) , \( -5 a^{3} + 7 a^{2} + 29 a\) , \( -2 a^{3} + 2 a^{2} + 12 a + 3\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-12a-1\right){x}{y}+\left(2a^{3}-3a^{2}-11a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(-5a^{3}+7a^{2}+29a\right){x}-2a^{3}+2a^{2}+12a+3$ |
| 15.1-b2 |
15.1-b |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{18} \cdot 5^{6} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$208.0024062$ |
1.550832337 |
\( \frac{7523340077821071614}{6053445140625} a^{3} - \frac{1472907287154167804}{672605015625} a^{2} - \frac{8511741355927900849}{1210689028125} a + \frac{9870567943499905918}{6053445140625} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( -a - 1\) , \( a^{3} - 2 a^{2} - 5 a + 2\) , \( -116 a^{3} + 198 a^{2} + 666 a - 113\) , \( 766 a^{3} - 1414 a^{2} - 4234 a + 1389\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+\left(a^{3}-2a^{2}-5a+2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-116a^{3}+198a^{2}+666a-113\right){x}+766a^{3}-1414a^{2}-4234a+1389$ |
| 15.1-c1 |
15.1-c |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{36} \cdot 5^{7} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$3.406718831$ |
$9.862879875$ |
3.507232522 |
\( -\frac{31432147885498162672657891129748}{11726143382578056328125} a^{3} + \frac{2725994792062218120973240077578}{1302904820286450703125} a^{2} + \frac{45085654006815651289450185018643}{2345228676515611265625} a + \frac{143740395124152286345601952770924}{11726143382578056328125} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( a^{3} - a^{2} - 7 a - 1\) , \( a^{3} - 2 a^{2} - 5 a + 2\) , \( 935 a^{3} - 2619 a^{2} - 1861 a + 518\) , \( -7527 a^{3} + 17643 a^{2} + 13079 a - 3620\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+\left(a^{3}-2a^{2}-5a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-7a-1\right){x}^{2}+\left(935a^{3}-2619a^{2}-1861a+518\right){x}-7527a^{3}+17643a^{2}+13079a-3620$ |
| 15.1-c2 |
15.1-c |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{18} \cdot 5^{14} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 7 \) |
$1.703359415$ |
$39.45151950$ |
3.507232522 |
\( -\frac{76967522773882105931176}{2364627008056640625} a^{3} + \frac{14387115873474141640261}{262736334228515625} a^{2} + \frac{81864909371396191405916}{472925401611328125} a + \frac{210874238481835997483188}{2364627008056640625} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{2} + 3 a + 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 82 a^{3} - 175 a^{2} - 304 a - 97\) , \( 342 a^{3} - 1194 a^{2} - 69 a + 791\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{2}+3a+2\right){x}^{2}+\left(82a^{3}-175a^{2}-304a-97\right){x}+342a^{3}-1194a^{2}-69a+791$ |
| 15.1-c3 |
15.1-c |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{9} \cdot 5^{28} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$3.406718831$ |
$4.931439937$ |
3.507232522 |
\( \frac{47648171056659673170365029753772}{733248889446258544921875} a^{3} + \frac{12380332503815016740414494474858}{81472098827362060546875} a^{2} + \frac{7702446558294286004513908740323}{146649777889251708984375} a - \frac{14264416385566401850172792772836}{733248889446258544921875} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{2} + 3 a + 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 27 a^{3} - 70 a^{2} - 104 a - 32\) , \( 780 a^{3} - 2883 a^{2} + 55 a + 2062\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{2}+3a+2\right){x}^{2}+\left(27a^{3}-70a^{2}-104a-32\right){x}+780a^{3}-2883a^{2}+55a+2062$ |
| 15.1-c4 |
15.1-c |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{9} \cdot 5^{7} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 7 \) |
$0.851679707$ |
$78.90303900$ |
3.507232522 |
\( -\frac{2923900801089717721}{1537734375} a^{3} + \frac{908161240961771881}{170859375} a^{2} + \frac{1158550046358000761}{307546875} a - \frac{1628555100466273727}{1537734375} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{2} + 3 a + 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 57 a^{3} - 155 a^{2} - 124 a + 18\) , \( 465 a^{3} - 1293 a^{2} - 939 a + 240\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{2}+3a+2\right){x}^{2}+\left(57a^{3}-155a^{2}-124a+18\right){x}+465a^{3}-1293a^{2}-939a+240$ |
| 15.1-d1 |
15.1-d |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{2} \cdot 5^{6} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$123.9995510$ |
2.773561860 |
\( \frac{49544357113354172279}{140625} a^{3} - \frac{9701127892841205619}{15625} a^{2} - \frac{56051453369519854864}{28125} a + \frac{64996470194594890648}{140625} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 12 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( -207 a^{3} + 352 a^{2} + 1150 a - 266\) , \( -2527 a^{3} + 4545 a^{2} + 14439 a - 3352\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-12a-1\right){x}{y}+\left(a^{3}-2a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(-207a^{3}+352a^{2}+1150a-266\right){x}-2527a^{3}+4545a^{2}+14439a-3352$ |
| 15.1-d2 |
15.1-d |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{3} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$495.9982043$ |
2.773561860 |
\( \frac{603206669}{375} a^{3} + \frac{5193612273}{125} a^{2} + \frac{4567351946}{75} a - \frac{5729111297}{375} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 12 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( -17 a^{3} + 17 a^{2} + 75 a - 16\) , \( -31 a^{3} + 145 a^{2} + 318 a - 76\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-12a-1\right){x}{y}+\left(a^{3}-2a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(-17a^{3}+17a^{2}+75a-16\right){x}-31a^{3}+145a^{2}+318a-76$ |
| 15.1-d3 |
15.1-d |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5 \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$495.9982043$ |
2.773561860 |
\( -\frac{3838108758152641}{135} a^{3} + \frac{333973895650741}{15} a^{2} + \frac{5500652138240714}{27} a + \frac{17518288921560178}{135} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( 2 a^{3} - 3 a^{2} - 11 a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 71 a^{3} - 205 a^{2} - 154 a + 44\) , \( -760 a^{3} + 2119 a^{2} + 1501 a - 414\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-11a-2\right){x}^{2}+\left(71a^{3}-205a^{2}-154a+44\right){x}-760a^{3}+2119a^{2}+1501a-414$ |
| 15.1-d4 |
15.1-d |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{6} \cdot 5^{2} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$123.9995510$ |
2.773561860 |
\( -\frac{153034061914417952797201441}{18225} a^{3} + \frac{47532258842630578293674551}{2025} a^{2} + \frac{60637342903067714951121296}{3645} a - \frac{85237087782169883585842667}{18225} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( 2 a^{3} - 3 a^{2} - 11 a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 1126 a^{3} - 3110 a^{2} - 2219 a + 624\) , \( -47333 a^{3} + 132465 a^{2} + 93858 a - 26381\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-11a-2\right){x}^{2}+\left(1126a^{3}-3110a^{2}-2219a+624\right){x}-47333a^{3}+132465a^{2}+93858a-26381$ |
| 15.1-e1 |
15.1-e |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{7} \) |
$16.81332$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.041918457$ |
$146.4241777$ |
2.562728516 |
\( \frac{10776641562922778}{373669453125} a^{3} - \frac{3376708614895658}{41518828125} a^{2} - \frac{4028690548195123}{74733890625} a + \frac{5573198186123311}{373669453125} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 2\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( 0\) , \( -22 a^{3} + 40 a^{2} + 124 a - 32\) , \( 81 a^{3} - 140 a^{2} - 461 a + 92\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-2\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-5a+1\right){x}^{2}+\left(-22a^{3}+40a^{2}+124a-32\right){x}+81a^{3}-140a^{2}-461a+92$ |
| 17.1-a1 |
17.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$17.07844$ |
$(-2a^3+3a^2+11a+4)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
|
\( 1 \) |
$1$ |
$116.5933393$ |
3.607076940 |
\( -\frac{204800}{17} a^{3} - \frac{507904}{17} a^{2} - \frac{225280}{17} a + \frac{53248}{17} \) |
\( \bigl[0\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 6 a - 1\) , \( 2 a^{3} - 3 a^{2} - 12 a + 3\) , \( a^{3} - 2 a^{2} - 6 a\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(2a^{3}-3a^{2}-12a+3\right){x}+a^{3}-2a^{2}-6a$ |
| 17.1-b1 |
17.1-b |
$2$ |
$3$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$17.07844$ |
$(-2a^3+3a^2+11a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$13.06444747$ |
0.876657687 |
\( \frac{1852104612561341327212}{4913} a^{3} - \frac{3263894094203014292758}{4913} a^{2} - \frac{10476788939515850994524}{4913} a + \frac{2429747168736670660959}{4913} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a + 1\) , \( -a^{3} + 2 a^{2} + 5 a\) , \( 2 a^{3} - 3 a^{2} - 12 a\) , \( -79 a^{3} + 216 a^{2} + 167 a - 44\) , \( 819 a^{3} - 2296 a^{2} - 1615 a + 455\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a+1\right){x}{y}+\left(2a^{3}-3a^{2}-12a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a\right){x}^{2}+\left(-79a^{3}+216a^{2}+167a-44\right){x}+819a^{3}-2296a^{2}-1615a+455$ |
| 17.1-b2 |
17.1-b |
$2$ |
$3$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$17.07844$ |
$(-2a^3+3a^2+11a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1058.220245$ |
0.876657687 |
\( \frac{5290407}{17} a^{3} - \frac{8798623}{17} a^{2} - \frac{30728361}{17} a + \frac{3791108}{17} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a + 1\) , \( -a^{3} + 2 a^{2} + 5 a\) , \( 2 a^{3} - 3 a^{2} - 12 a\) , \( 6 a^{3} - 19 a^{2} - 3 a + 1\) , \( -44 a^{3} + 121 a^{2} + 95 a - 28\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a+1\right){x}{y}+\left(2a^{3}-3a^{2}-12a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a\right){x}^{2}+\left(6a^{3}-19a^{2}-3a+1\right){x}-44a^{3}+121a^{2}+95a-28$ |
| 17.2-a1 |
17.2-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( - 17^{7} \) |
$17.07844$ |
$(a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.020176703$ |
$525.5172548$ |
2.213562094 |
\( -\frac{342377720046516}{410338673} a^{3} + \frac{603126668896495}{410338673} a^{2} + \frac{1937308239909663}{410338673} a - \frac{448146690701028}{410338673} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 12 a - 1\) , \( -a^{3} + 2 a^{2} + 6 a - 1\) , \( 2 a^{3} - 3 a^{2} - 12 a\) , \( 50 a^{3} - 142 a^{2} - 88 a + 24\) , \( -660 a^{3} + 1844 a^{2} + 1314 a - 371\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-12a-1\right){x}{y}+\left(2a^{3}-3a^{2}-12a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+6a-1\right){x}^{2}+\left(50a^{3}-142a^{2}-88a+24\right){x}-660a^{3}+1844a^{2}+1314a-371$ |
| 23.1-a1 |
23.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$17.73610$ |
$(a^3-a^2-6a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$59.11900908$ |
0.440781780 |
\( \frac{14085956632117}{23} a^{3} + \frac{32944320140214}{23} a^{2} + \frac{11392997799081}{23} a - \frac{4218860460304}{23} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 12 a - 1\) , \( a^{3} - a^{2} - 6 a - 1\) , \( 2 a^{3} - 3 a^{2} - 11 a\) , \( 3 a^{3} - 6 a^{2} - 28 a + 2\) , \( -105 a^{3} + 282 a^{2} + 169 a - 51\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-12a-1\right){x}{y}+\left(2a^{3}-3a^{2}-11a\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(3a^{3}-6a^{2}-28a+2\right){x}-105a^{3}+282a^{2}+169a-51$ |
| 23.1-b1 |
23.1-b |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( - 23^{3} \) |
$17.73610$ |
$(a^3-a^2-6a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$309.5534733$ |
2.307980687 |
\( -\frac{2845083312128}{12167} a^{3} + \frac{2222438174720}{12167} a^{2} + \frac{20401968717824}{12167} a + \frac{13000207810560}{12167} \) |
\( \bigl[0\) , \( -a^{2} + a + 2\) , \( a\) , \( a^{2} + 2 a + 1\) , \( -a^{2} - 3 a + 1\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(a^{2}+2a+1\right){x}-a^{2}-3a+1$ |
| 25.1-a1 |
25.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{2} \) |
$17.92192$ |
$(-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$349.1260861$ |
2.603027695 |
\( -171827 a^{3} + 486724 a^{2} + 319388 a - 113766 \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( 2 a^{3} - 3 a^{2} - 12 a\) , \( 2 a^{3} - 2 a^{2} - 10 a + 2\) , \( 2 a^{2} + a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(2a^{3}-3a^{2}-12a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(2a^{3}-2a^{2}-10a+2\right){x}+2a^{2}+a-5$ |
| 25.1-b1 |
25.1-b |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{8} \) |
$17.92192$ |
$(-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.119947467$ |
$1147.926625$ |
4.106404182 |
\( -171827 a^{3} + 486724 a^{2} + 319388 a - 113766 \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( -a^{3} + 2 a^{2} + 6 a\) , \( 0\) , \( 7 a^{3} - 6 a^{2} - 46 a - 18\) , \( -8 a^{3} + 8 a^{2} + 58 a + 34\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}={x}^{3}+\left(-a^{3}+2a^{2}+6a\right){x}^{2}+\left(7a^{3}-6a^{2}-46a-18\right){x}-8a^{3}+8a^{2}+58a+34$ |
| 27.1-a1 |
27.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{3} \) |
$18.09516$ |
$(a^3-10a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$514.6598630$ |
3.837220793 |
\( -332 a^{3} + \frac{383}{3} a^{2} + \frac{8303}{3} a + \frac{5081}{3} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 11 a - 1\) , \( a^{3} - a^{2} - 8 a - 2\) , \( a^{3} - a^{2} - 7 a - 2\) , \( 5 a^{3} - 7 a^{2} - 33 a + 1\) , \( 6 a^{3} - 10 a^{2} - 35 a + 5\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-11a-1\right){x}{y}+\left(a^{3}-a^{2}-7a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-8a-2\right){x}^{2}+\left(5a^{3}-7a^{2}-33a+1\right){x}+6a^{3}-10a^{2}-35a+5$ |
| 27.1-b1 |
27.1-b |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{9} \) |
$18.09516$ |
$(a^3-10a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$135.7034903$ |
1.011783300 |
\( -\frac{297144212}{27} a^{3} + \frac{523526792}{27} a^{2} + \frac{1681240447}{27} a - \frac{389849032}{27} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 12 a - 1\) , \( -2 a^{3} + 3 a^{2} + 12 a\) , \( 1\) , \( 6 a^{3} - 24 a^{2} + a + 15\) , \( -39 a^{3} + 102 a^{2} + 81 a - 15\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-12a-1\right){x}{y}+{y}={x}^{3}+\left(-2a^{3}+3a^{2}+12a\right){x}^{2}+\left(6a^{3}-24a^{2}+a+15\right){x}-39a^{3}+102a^{2}+81a-15$ |
| 31.1-a1 |
31.1-a |
$2$ |
$3$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31 \) |
$18.41036$ |
$(-a^2+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$364.4533344$ |
2.717305182 |
\( -\frac{15496}{31} a^{3} + \frac{12364}{31} a^{2} + \frac{144782}{31} a - \frac{44035}{31} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( 2 a^{3} - 3 a^{2} - 12 a - 2\) , \( 1\) , \( 2 a^{3} - 4 a^{2} - 9 a + 7\) , \( 2 a^{3} - 3 a^{2} - 11 a + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+{y}={x}^{3}+\left(2a^{3}-3a^{2}-12a-2\right){x}^{2}+\left(2a^{3}-4a^{2}-9a+7\right){x}+2a^{3}-3a^{2}-11a+1$ |
| 31.1-a2 |
31.1-a |
$2$ |
$3$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$18.41036$ |
$(-a^2+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$364.4533344$ |
2.717305182 |
\( \frac{4050046292697}{29791} a^{3} - \frac{3163698751118}{29791} a^{2} - \frac{29042703245608}{29791} a - \frac{18506067686051}{29791} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 5 a\) , \( 0\) , \( 29 a^{3} - 51 a^{2} - 165 a + 38\) , \( 68 a^{3} - 121 a^{2} - 386 a + 90\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-2a^{2}-5a\right){x}^{2}+\left(29a^{3}-51a^{2}-165a+38\right){x}+68a^{3}-121a^{2}-386a+90$ |
| 31.4-a1 |
31.4-a |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( -31 \) |
$18.41036$ |
$(-a^3+a^2+6a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1074.744584$ |
2.003280498 |
\( -\frac{2540749}{31} a^{3} + \frac{1964465}{31} a^{2} + \frac{18227506}{31} a + \frac{11750070}{31} \) |
\( \bigl[a^{3} - a^{2} - 7 a - 1\) , \( -2 a^{3} + 3 a^{2} + 13 a + 2\) , \( 2 a^{3} - 3 a^{2} - 11 a - 1\) , \( -2 a^{3} + 4 a^{2} + 11 a - 1\) , \( -a^{3} + 2 a^{2} + 6 a - 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-7a-1\right){x}{y}+\left(2a^{3}-3a^{2}-11a-1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+13a+2\right){x}^{2}+\left(-2a^{3}+4a^{2}+11a-1\right){x}-a^{3}+2a^{2}+6a-3$ |
| 31.4-a2 |
31.4-a |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
31.4 |
\( 31 \) |
\( - 31^{2} \) |
$18.41036$ |
$(-a^3+a^2+6a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$537.3722922$ |
2.003280498 |
\( -\frac{40404368803761}{961} a^{3} + \frac{31562203300895}{961} a^{2} + \frac{289738653112130}{961} a + \frac{184622433663823}{961} \) |
\( \bigl[1\) , \( a^{2} - 2 a - 4\) , \( a + 1\) , \( -6 a^{3} - 5 a^{2} + 5 a + 4\) , \( -26 a^{3} - 55 a^{2} - 21 a + 5\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(-6a^{3}-5a^{2}+5a+4\right){x}-26a^{3}-55a^{2}-21a+5$ |
| 33.1-a1 |
33.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3 \cdot 11 \) |
$18.55480$ |
$(-a^2-2a), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$148.5358195$ |
1.107459074 |
\( \frac{2282042}{33} a^{3} - \frac{2127475}{11} a^{2} - \frac{4516514}{33} a + \frac{1282000}{33} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 2\) , \( -a^{2} + 2 a + 3\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( -2 a^{3} + a^{2} + 16 a + 9\) , \( -2 a^{3} + a^{2} + 13 a + 6\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-2\right){x}{y}+\left(a^{3}-2a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-2a^{3}+a^{2}+16a+9\right){x}-2a^{3}+a^{2}+13a+6$ |
| 33.1-b1 |
33.1-b |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( 3^{3} \cdot 11 \) |
$18.55480$ |
$(-a^2-2a), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$161.8618373$ |
3.620447128 |
\( -\frac{627331858}{297} a^{3} + \frac{122829874}{33} a^{2} + \frac{3548707420}{297} a - \frac{822662903}{297} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{2} + 3 a + 4\) , \( a^{3} - a^{2} - 6 a - 2\) , \( -125 a^{3} + 218 a^{2} + 711 a - 150\) , \( -1178 a^{3} + 2073 a^{2} + 6668 a - 1528\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-6a-2\right){y}={x}^{3}+\left(-a^{2}+3a+4\right){x}^{2}+\left(-125a^{3}+218a^{2}+711a-150\right){x}-1178a^{3}+2073a^{2}+6668a-1528$ |
| 39.1-a1 |
39.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{6} \cdot 13^{3} \) |
$18.94633$ |
$(-a^2-2a), (a^2-3a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$1$ |
$47.57845182$ |
0.709474500 |
\( -\frac{38540967683}{1601613} a^{3} + \frac{7436417060}{177957} a^{2} + \frac{17107738364}{123201} a - \frac{51413054260}{1601613} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 11 a - 1\) , \( 1\) , \( a^{3} - a^{2} - 6 a - 1\) , \( 18 a^{3} - 49 a^{2} - 42 a + 13\) , \( 110 a^{3} - 306 a^{2} - 225 a + 64\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-11a-1\right){x}{y}+\left(a^{3}-a^{2}-6a-1\right){y}={x}^{3}+{x}^{2}+\left(18a^{3}-49a^{2}-42a+13\right){x}+110a^{3}-306a^{2}-225a+64$ |
| 39.1-b1 |
39.1-b |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{9} \cdot 13^{3} \) |
$18.94633$ |
$(-a^2-2a), (a^2-3a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$332.1641721$ |
2.476562405 |
\( \frac{354612924967}{43243551} a^{3} - \frac{57375266173}{4804839} a^{2} - \frac{212028166483}{3326427} a + \frac{618519376157}{43243551} \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -6 a^{3} + 11 a^{2} + 31 a - 7\) , \( -7 a^{3} + 12 a^{2} + 42 a - 15\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{2}+3a+2\right){x}^{2}+\left(-6a^{3}+11a^{2}+31a-7\right){x}-7a^{3}+12a^{2}+42a-15$ |
| 39.1-c1 |
39.1-c |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{10} \cdot 13 \) |
$18.94633$ |
$(-a^2-2a), (a^2-3a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$0.064673029$ |
$286.0296400$ |
5.516844258 |
\( \frac{175071597525404524}{767637} a^{3} - \frac{54377099954462407}{85293} a^{2} - \frac{26680527102378199}{59049} a + \frac{97511578397424359}{767637} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 12 a\) , \( a^{2} - 2 a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 18 a^{3} - 38 a^{2} - 117 a + 25\) , \( 252 a^{3} - 390 a^{2} - 1341 a + 308\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-12a\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(18a^{3}-38a^{2}-117a+25\right){x}+252a^{3}-390a^{2}-1341a+308$ |
| 45.1-a1 |
45.1-a |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{3} \cdot 5^{5} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$0.122141109$ |
$493.6485273$ |
4.495481284 |
\( \frac{104004317}{3125} a^{3} + \frac{932618767}{3125} a^{2} + \frac{209749603}{625} a - \frac{201047921}{3125} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 8 a + 3\) , \( 2 a^{3} - 3 a^{2} - 11 a - 1\) , \( 66 a^{3} - 192 a^{2} - 124 a + 45\) , \( 660 a^{3} - 1865 a^{2} - 1312 a + 370\bigr] \) |
${y}^2+{x}{y}+\left(2a^{3}-3a^{2}-11a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+8a+3\right){x}^{2}+\left(66a^{3}-192a^{2}-124a+45\right){x}+660a^{3}-1865a^{2}-1312a+370$ |
| 45.1-a2 |
45.1-a |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{3} \cdot 5^{10} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.061070554$ |
$493.6485273$ |
4.495481284 |
\( \frac{60989446953572296}{9765625} a^{3} - \frac{108301460021125479}{9765625} a^{2} - \frac{68745971412015836}{1953125} a + \frac{84938930369240502}{9765625} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 8 a + 3\) , \( 2 a^{3} - 3 a^{2} - 11 a - 1\) , \( 121 a^{3} - 342 a^{2} - 244 a + 65\) , \( -625 a^{3} + 1730 a^{2} + 1233 a - 348\bigr] \) |
${y}^2+{x}{y}+\left(2a^{3}-3a^{2}-11a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+8a+3\right){x}^{2}+\left(121a^{3}-342a^{2}-244a+65\right){x}-625a^{3}+1730a^{2}+1233a-348$ |
| 45.1-b1 |
45.1-b |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{15} \cdot 5^{28} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \cdot 7 \) |
$1$ |
$1.800702187$ |
3.007366790 |
\( \frac{47648171056659673170365029753772}{733248889446258544921875} a^{3} + \frac{12380332503815016740414494474858}{81472098827362060546875} a^{2} + \frac{7702446558294286004513908740323}{146649777889251708984375} a - \frac{14264416385566401850172792772836}{733248889446258544921875} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 2\) , \( a + 1\) , \( 0\) , \( 3861 a^{3} - 10794 a^{2} - 7647 a + 2148\) , \( -2870912 a^{3} + 8025347 a^{2} + 5687689 a - 1599122\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-2\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3861a^{3}-10794a^{2}-7647a+2148\right){x}-2870912a^{3}+8025347a^{2}+5687689a-1599122$ |
| 45.1-b2 |
45.1-b |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{42} \cdot 5^{7} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$3.601404375$ |
3.007366790 |
\( -\frac{31432147885498162672657891129748}{11726143382578056328125} a^{3} + \frac{2725994792062218120973240077578}{1302904820286450703125} a^{2} + \frac{45085654006815651289450185018643}{2345228676515611265625} a + \frac{143740395124152286345601952770924}{11726143382578056328125} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 11 a\) , \( -a^{3} + a^{2} + 7 a + 2\) , \( a^{3} - a^{2} - 7 a - 1\) , \( -56317 a^{3} + 99285 a^{2} + 318506 a - 74119\) , \( 10386254 a^{3} - 18303002 a^{2} - 58752378 a + 13623661\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-11a\right){x}{y}+\left(a^{3}-a^{2}-7a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+7a+2\right){x}^{2}+\left(-56317a^{3}+99285a^{2}+318506a-74119\right){x}+10386254a^{3}-18303002a^{2}-58752378a+13623661$ |
| 45.1-b3 |
45.1-b |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{24} \cdot 5^{14} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \cdot 7 \) |
$1$ |
$28.81123500$ |
3.007366790 |
\( -\frac{76967522773882105931176}{2364627008056640625} a^{3} + \frac{14387115873474141640261}{262736334228515625} a^{2} + \frac{81864909371396191405916}{472925401611328125} a + \frac{210874238481835997483188}{2364627008056640625} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 11 a\) , \( -a^{3} + a^{2} + 7 a + 2\) , \( a^{3} - a^{2} - 7 a - 1\) , \( -3842 a^{3} + 6770 a^{2} + 21726 a - 5049\) , \( 124699 a^{3} - 219754 a^{2} - 705410 a + 163548\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-11a\right){x}{y}+\left(a^{3}-a^{2}-7a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+7a+2\right){x}^{2}+\left(-3842a^{3}+6770a^{2}+21726a-5049\right){x}+124699a^{3}-219754a^{2}-705410a+163548$ |
| 45.1-b4 |
45.1-b |
$4$ |
$4$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{15} \cdot 5^{7} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$0 \le r \le 2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$230.4898800$ |
3.007366790 |
\( -\frac{2923900801089717721}{1537734375} a^{3} + \frac{908161240961771881}{170859375} a^{2} + \frac{1158550046358000761}{307546875} a - \frac{1628555100466273727}{1537734375} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a + 2\) , \( -a^{3} + 2 a^{2} + 6 a - 2\) , \( a^{3} - 2 a^{2} - 5 a + 2\) , \( -26 a^{3} - 63 a^{2} - 26 a + 5\) , \( 165 a^{3} + 382 a^{2} + 128 a - 47\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a+2\right){x}{y}+\left(a^{3}-2a^{2}-5a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+6a-2\right){x}^{2}+\left(-26a^{3}-63a^{2}-26a+5\right){x}+165a^{3}+382a^{2}+128a-47$ |
| 45.1-c1 |
45.1-c |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5 \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$243.6351081$ |
3.633008012 |
\( -\frac{17638}{45} a^{3} - \frac{5397}{5} a^{2} - \frac{6598}{9} a - \frac{4736}{45} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( -a^{3} + a^{2} + 8 a + 1\) , \( a + 1\) , \( 0\) , \( -a^{3} + 2 a^{2} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+8a+1\right){x}^{2}-a^{3}+2a^{2}+4a+1$ |
| 45.1-d1 |
45.1-d |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{15} \cdot 5^{3} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$1.155891761$ |
$121.7352147$ |
4.196525582 |
\( -\frac{5160440524}{2460375} a^{3} + \frac{1069321939}{273375} a^{2} + \frac{8607014084}{492075} a + \frac{20961784912}{2460375} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 6 a\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 3 a^{3} - 5 a^{2} - 18 a - 6\) , \( 2 a^{3} - 3 a^{2} - 12 a - 5\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-6a\right){x}^{2}+\left(3a^{3}-5a^{2}-18a-6\right){x}+2a^{3}-3a^{2}-12a-5$ |
| 45.1-d2 |
45.1-d |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{24} \cdot 5^{6} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{3} \) |
$2.311783522$ |
$30.43380369$ |
4.196525582 |
\( \frac{7523340077821071614}{6053445140625} a^{3} - \frac{1472907287154167804}{672605015625} a^{2} - \frac{8511741355927900849}{1210689028125} a + \frac{9870567943499905918}{6053445140625} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 6 a\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -2 a^{3} + 12 a + 9\) , \( 10 a^{3} - 4 a^{2} - 66 a - 48\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-6a\right){x}^{2}+\left(-2a^{3}+12a+9\right){x}+10a^{3}-4a^{2}-66a-48$ |
| 45.1-e1 |
45.1-e |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{20} \cdot 5^{7} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$17.23701152$ |
1.799229224 |
\( \frac{10776641562922778}{373669453125} a^{3} - \frac{3376708614895658}{41518828125} a^{2} - \frac{4028690548195123}{74733890625} a + \frac{5573198186123311}{373669453125} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 11 a\) , \( 0\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( 10 a^{3} + 5 a^{2} - 18 a + 3\) , \( 10 a^{3} + 3 a^{2} - 20 a + 5\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-11a\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(10a^{3}+5a^{2}-18a+3\right){x}+10a^{3}+3a^{2}-20a+5$ |
| 45.1-f1 |
45.1-f |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{9} \cdot 5^{10} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1.297251928$ |
$21.42180451$ |
4.143877051 |
\( \frac{60989446953572296}{9765625} a^{3} - \frac{108301460021125479}{9765625} a^{2} - \frac{68745971412015836}{1953125} a + \frac{84938930369240502}{9765625} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( -a^{3} + 2 a^{2} + 5 a - 1\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( -447 a^{3} + 773 a^{2} + 2504 a - 581\) , \( -7097 a^{3} + 12401 a^{2} + 39979 a - 9270\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+\left(a^{3}-2a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-1\right){x}^{2}+\left(-447a^{3}+773a^{2}+2504a-581\right){x}-7097a^{3}+12401a^{2}+39979a-9270$ |
| 45.1-f2 |
45.1-f |
$2$ |
$2$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{9} \cdot 5^{5} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$0.648625964$ |
$85.68721805$ |
4.143877051 |
\( \frac{104004317}{3125} a^{3} + \frac{932618767}{3125} a^{2} + \frac{209749603}{625} a - \frac{201047921}{3125} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( -a^{2} + a + 2\) , \( 2 a^{3} - 3 a^{2} - 12 a\) , \( -78 a^{3} + 143 a^{2} + 433 a - 134\) , \( -628 a^{3} + 1123 a^{2} + 3527 a - 922\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(2a^{3}-3a^{2}-12a\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-78a^{3}+143a^{2}+433a-134\right){x}-628a^{3}+1123a^{2}+3527a-922$ |
| 45.1-g1 |
45.1-g |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{3} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$652.4845438$ |
0.810803224 |
\( \frac{603206669}{375} a^{3} + \frac{5193612273}{125} a^{2} + \frac{4567351946}{75} a - \frac{5729111297}{375} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 11 a - 1\) , \( -2 a^{3} + 3 a^{2} + 12 a + 1\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -74 a^{3} + 133 a^{2} + 415 a - 116\) , \( 542 a^{3} - 932 a^{2} - 3101 a + 569\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-11a-1\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+12a+1\right){x}^{2}+\left(-74a^{3}+133a^{2}+415a-116\right){x}+542a^{3}-932a^{2}-3101a+569$ |
| 45.1-g2 |
45.1-g |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{8} \cdot 5^{6} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$163.1211359$ |
0.810803224 |
\( \frac{49544357113354172279}{140625} a^{3} - \frac{9701127892841205619}{15625} a^{2} - \frac{56051453369519854864}{28125} a + \frac{64996470194594890648}{140625} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( 2 a^{3} - 3 a^{2} - 13 a - 2\) , \( 2 a^{3} - 3 a^{2} - 12 a - 1\) , \( 21 a^{3} - 20 a^{2} - 167 a - 114\) , \( -95 a^{3} + 103 a^{2} + 755 a + 466\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(2a^{3}-3a^{2}-12a-1\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-13a-2\right){x}^{2}+\left(21a^{3}-20a^{2}-167a-114\right){x}-95a^{3}+103a^{2}+755a+466$ |
| 45.1-g3 |
45.1-g |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{12} \cdot 5^{2} \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$1$ |
$6.041523553$ |
0.810803224 |
\( -\frac{153034061914417952797201441}{18225} a^{3} + \frac{47532258842630578293674551}{2025} a^{2} + \frac{60637342903067714951121296}{3645} a - \frac{85237087782169883585842667}{18225} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a + 2\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( 1\) , \( -31365 a^{3} + 55306 a^{2} + 177373 a - 41331\) , \( 4301450 a^{3} - 7580107 a^{2} - 24332259 a + 5641952\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+4a\right){x}^{2}+\left(-31365a^{3}+55306a^{2}+177373a-41331\right){x}+4301450a^{3}-7580107a^{2}-24332259a+5641952$ |
| 45.1-g4 |
45.1-g |
$4$ |
$6$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{9} \cdot 5 \) |
$19.28829$ |
$(-a^2-2a), (-a^2-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$24.16609421$ |
0.810803224 |
\( -\frac{3838108758152641}{135} a^{3} + \frac{333973895650741}{15} a^{2} + \frac{5500652138240714}{27} a + \frac{17518288921560178}{135} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a + 2\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( 1\) , \( -570 a^{3} + 1036 a^{2} + 3173 a - 931\) , \( 159531 a^{3} - 280959 a^{2} - 902691 a + 208233\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+4a\right){x}^{2}+\left(-570a^{3}+1036a^{2}+3173a-931\right){x}+159531a^{3}-280959a^{2}-902691a+208233$ |
| 48.1-a1 |
48.1-a |
$1$ |
$1$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{9} \) |
$19.44452$ |
$(-a^2-2a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$0.087522819$ |
$230.5622088$ |
5.416371122 |
\( \frac{75135705029273}{19683} a^{3} - \frac{23339235334832}{2187} a^{2} - \frac{148788039174260}{19683} a + \frac{83670089134901}{39366} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( -2 a^{3} + 3 a^{2} + 12 a + 2\) , \( 2 a^{3} - 3 a^{2} - 12 a - 1\) , \( -53 a^{3} + 100 a^{2} + 293 a - 102\) , \( 437 a^{3} - 730 a^{2} - 2531 a + 338\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+\left(2a^{3}-3a^{2}-12a-1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+12a+2\right){x}^{2}+\left(-53a^{3}+100a^{2}+293a-102\right){x}+437a^{3}-730a^{2}-2531a+338$ |
| 48.1-b1 |
48.1-b |
$2$ |
$3$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{3} \) |
$19.44452$ |
$(-a^2-2a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.257311926$ |
$247.9307815$ |
5.707791499 |
\( -\frac{188807}{54} a^{3} + \frac{21622}{3} a^{2} + \frac{483889}{27} a - \frac{595309}{54} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a + 1\) , \( a^{3} - 2 a^{2} - 6 a + 1\) , \( 2 a^{3} - 3 a^{2} - 12 a - 1\) , \( -48 a^{3} + 85 a^{2} + 272 a - 64\) , \( -393 a^{3} + 693 a^{2} + 2222 a - 519\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a+1\right){x}{y}+\left(2a^{3}-3a^{2}-12a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-6a+1\right){x}^{2}+\left(-48a^{3}+85a^{2}+272a-64\right){x}-393a^{3}+693a^{2}+2222a-519$ |
| 48.1-b2 |
48.1-b |
$2$ |
$3$ |
4.4.17989.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{12} \cdot 3^{9} \) |
$19.44452$ |
$(-a^2-2a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$0.085770642$ |
$247.9307815$ |
5.707791499 |
\( \frac{993924599}{19683} a^{3} - \frac{2470940827}{17496} a^{2} - \frac{3940446391}{39366} a + \frac{2345944507}{78732} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 8 a - 1\) , \( a + 1\) , \( 9 a^{3} - 24 a^{2} - 23 a + 8\) , \( -16 a^{3} + 46 a^{2} + 26 a - 9\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-8a-1\right){x}^{2}+\left(9a^{3}-24a^{2}-23a+8\right){x}-16a^{3}+46a^{2}+26a-9$ |
*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.
