| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{5} \) |
$14.24610$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.044341762$ |
$376.5317480$ |
2.561287979 |
\( \frac{2539168}{3125} a^{3} - \frac{38097}{625} a^{2} - \frac{12376558}{3125} a - \frac{96508}{3125} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{2} + a - 3\) , \( -a^{3} + a^{2}\) , \( -a^{3} + 2 a^{2} + a - 6\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-a^{3}+a^{2}\right){x}-a^{3}+2a^{2}+a-6$ |
| 5.2-a1 |
5.2-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{2} \) |
$14.24610$ |
$(-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.036411483$ |
$1317.379880$ |
2.943426863 |
\( \frac{8133357}{25} a^{3} + \frac{7223444}{25} a^{2} - \frac{42380619}{25} a - \frac{9156051}{5} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 3 a + 1\) , \( a + 1\) , \( -a^{3} + 2 a\) , \( -a\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-a^{3}+2a\right){x}-a$ |
| 5.2-b1 |
5.2-b |
$2$ |
$5$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{10} \) |
$14.24610$ |
$(-a^2-a+1)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$2.708762032$ |
$86.84611140$ |
2.887055124 |
\( \frac{3364042267}{9765625} a^{3} + \frac{1964925889}{9765625} a^{2} - \frac{25655280414}{9765625} a - \frac{3815938726}{1953125} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -5 a^{3} - 12 a^{2} + a + 10\) , \( -30 a^{3} - 71 a^{2} + 13 a + 63\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-5a^{3}-12a^{2}+a+10\right){x}-30a^{3}-71a^{2}+13a+63$ |
| 5.2-b2 |
5.2-b |
$2$ |
$5$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{2} \) |
$14.24610$ |
$(-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$13.54381016$ |
$0.138953778$ |
2.887055124 |
\( \frac{5357496973442499987}{25} a^{3} + \frac{4190018497495156254}{25} a^{2} - \frac{29759807723130703029}{25} a - \frac{6303903372448402516}{5} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{2} + a - 3\) , \( a\) , \( -1147 a^{3} + 1237 a^{2} + 5259 a - 4929\) , \( -37920 a^{3} + 42325 a^{2} + 176448 a - 163829\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-1147a^{3}+1237a^{2}+5259a-4929\right){x}-37920a^{3}+42325a^{2}+176448a-163829$ |
| 7.1-a1 |
7.1-a |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{5} \) |
$14.85805$ |
$(-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$625.1170218$ |
1.198712947 |
\( \frac{224443276}{16807} a^{3} + \frac{251959446}{16807} a^{2} - \frac{427552997}{16807} a + \frac{85530617}{16807} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a^{2} - 3\) , \( 2 a^{3} - 6 a^{2} - 16 a + 12\) , \( -a^{3} - 9 a^{2} - 8 a + 13\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(2a^{3}-6a^{2}-16a+12\right){x}-a^{3}-9a^{2}-8a+13$ |
| 7.1-a2 |
7.1-a |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{10} \) |
$14.85805$ |
$(-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$78.13962772$ |
1.198712947 |
\( \frac{500718065734107078}{282475249} a^{3} + \frac{1176400322528301747}{282475249} a^{2} - \frac{240441686104387112}{282475249} a - \frac{1065616788500768418}{282475249} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -59 a^{3} - 68 a^{2} + 316 a + 398\) , \( 750 a^{3} + 650 a^{2} - 3904 a - 4142\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+3\right){x}^{2}+\left(-59a^{3}-68a^{2}+316a+398\right){x}+750a^{3}+650a^{2}-3904a-4142$ |
| 23.1-a1 |
23.1-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( - 23^{5} \) |
$17.24014$ |
$(a^2+2a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$400.8912646$ |
3.074966976 |
\( -\frac{66886133}{12167} a^{3} - \frac{146500454}{12167} a^{2} + \frac{46547253}{12167} a + \frac{145466034}{12167} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 4 a + 5\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -4 a^{3} + 10 a^{2} + a - 7\) , \( 2 a^{3} - 3 a^{2} - 7 a + 7\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+5\right){x}^{2}+\left(-4a^{3}+10a^{2}+a-7\right){x}+2a^{3}-3a^{2}-7a+7$ |
| 23.1-b1 |
23.1-b |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$17.24014$ |
$(a^2+2a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.413046127$ |
$302.5170590$ |
3.833736665 |
\( -\frac{135438}{23} a^{3} - \frac{62386}{23} a^{2} + \frac{855483}{23} a + \frac{93087}{23} \) |
\( \bigl[a\) , \( -a^{3} + 5 a + 1\) , \( a\) , \( a^{3} - 5 a^{2} + a + 10\) , \( 3 a^{3} - 8 a^{2} - 3 a + 11\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(a^{3}-5a^{2}+a+10\right){x}+3a^{3}-8a^{2}-3a+11$ |
| 25.1-a1 |
25.1-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{12} \) |
$17.42076$ |
$(-a), (-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 11 \) |
$0.143056813$ |
$94.14464040$ |
4.545385136 |
\( -\frac{406578548314}{48828125} a^{3} + \frac{471762130587}{48828125} a^{2} + \frac{1902320810713}{48828125} a - \frac{361462195458}{9765625} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a^{3} + 3 a\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 3 a^{2} - 8\) , \( -3 a^{3} + a^{2} + 9 a + 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(3a^{2}-8\right){x}-3a^{3}+a^{2}+9a+3$ |
| 25.1-b1 |
25.1-b |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{15} \) |
$17.42076$ |
$(-a), (-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$0.027295642$ |
$164.6213356$ |
3.584483980 |
\( \frac{731061055982}{1220703125} a^{3} - \frac{229137199378}{244140625} a^{2} - \frac{3913870618717}{1220703125} a + \frac{5638620853733}{1220703125} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} + a^{2} + 2 a + 1\) , \( -a^{3} + 3 a^{2} + 3 a - 6\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-a^{3}+a^{2}+2a+1\right){x}-a^{3}+3a^{2}+3a-6$ |
| 25.1-c1 |
25.1-c |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{16} \) |
$17.42076$ |
$(-a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$0.216356990$ |
$75.63985294$ |
3.514742414 |
\( -\frac{3331278739269177}{6103515625} a^{3} - \frac{2743553837201984}{6103515625} a^{2} + \frac{17099165551806634}{6103515625} a + \frac{3636486596530956}{1220703125} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 3 a + 1\) , \( a^{2} + a - 3\) , \( 90 a^{3} + 216 a^{2} - 38 a - 197\) , \( -92 a^{3} - 215 a^{2} + 46 a + 194\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(90a^{3}+216a^{2}-38a-197\right){x}-92a^{3}-215a^{2}+46a+194$ |
| 25.1-c2 |
25.1-c |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.42076$ |
$(-a), (-a^2-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 7 \) |
$0.432713981$ |
$151.2797058$ |
3.514742414 |
\( \frac{1466338053}{78125} a^{3} - \frac{1630454449}{78125} a^{2} - \frac{6858147526}{78125} a + \frac{1246449766}{15625} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 6\) , \( a\) , \( 3 a^{3} - 2 a^{2} - 12 a + 10\) , \( 6 a^{3} - 8 a^{2} - 16 a + 15\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+6\right){x}^{2}+\left(3a^{3}-2a^{2}-12a+10\right){x}+6a^{3}-8a^{2}-16a+15$ |
| 25.1-d1 |
25.1-d |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{5} \) |
$17.42076$ |
$(-a), (-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.016892733$ |
$1519.402522$ |
3.149971396 |
\( -\frac{21090746}{625} a^{3} - \frac{56533982}{625} a^{2} + \frac{1626807}{625} a + \frac{11486313}{125} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 4 a\) , \( a^{2} + a - 2\) , \( 3 a^{3} + a^{2} - 17 a - 13\) , \( -9 a^{3} - 7 a^{2} + 45 a + 46\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(3a^{3}+a^{2}-17a-13\right){x}-9a^{3}-7a^{2}+45a+46$ |
| 25.2-a1 |
25.2-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{2} \) |
$17.42076$ |
$(a^3-a^2-3a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$392.5605785$ |
3.011067891 |
\( -297245056 a^{3} - \frac{1320357839}{5} a^{2} + \frac{7744841263}{5} a + 1673226125 \) |
\( \bigl[a + 1\) , \( -a^{3} + 4 a + 2\) , \( a^{3} - 3 a\) , \( -a^{3} - 4 a^{2} + 3 a + 14\) , \( -4 a^{3} + a^{2} + 18 a - 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+4a+2\right){x}^{2}+\left(-a^{3}-4a^{2}+3a+14\right){x}-4a^{3}+a^{2}+18a-8$ |
| 25.2-b1 |
25.2-b |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.42076$ |
$(a^3-a^2-3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$4.040627552$ |
$67.99894672$ |
4.214973660 |
\( -\frac{50461999454}{25} a^{3} - \frac{39517290421}{25} a^{2} + \frac{256560530658}{25} a + \frac{53953629767}{5} \) |
\( \bigl[a^{3} - 3 a\) , \( -a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 80 a^{3} + 71 a^{2} - 414 a - 447\) , \( 783 a^{3} + 696 a^{2} - 4078 a - 4409\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(80a^{3}+71a^{2}-414a-447\right){x}+783a^{3}+696a^{2}-4078a-4409$ |
| 25.2-b2 |
25.2-b |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{2} \) |
$17.42076$ |
$(a^3-a^2-3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.020313776$ |
$271.9957869$ |
4.214973660 |
\( \frac{87796}{5} a^{3} + 8260 a^{2} - \frac{410979}{5} a - 76388 \) |
\( \bigl[a^{3} - 3 a\) , \( -a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 5 a^{3} + 6 a^{2} - 24 a - 32\) , \( 11 a^{3} + 11 a^{2} - 56 a - 66\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a^{3}+6a^{2}-24a-32\right){x}+11a^{3}+11a^{2}-56a-66$ |
| 25.2-c1 |
25.2-c |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{10} \) |
$17.42076$ |
$(a^3-a^2-3a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$72.31565486$ |
0.554684699 |
\( -\frac{365938}{3125} a^{3} + \frac{39661}{625} a^{2} + \frac{454002}{3125} a - \frac{56742}{625} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( a^{3} - 4 a - 1\) , \( -3 a^{3} + 6 a^{2} + 4 a - 5\) , \( -33 a^{3} + 43 a^{2} + 140 a - 139\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(-3a^{3}+6a^{2}+4a-5\right){x}-33a^{3}+43a^{2}+140a-139$ |
| 25.2-d1 |
25.2-d |
$2$ |
$5$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{2} \) |
$17.42076$ |
$(a^3-a^2-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$8.177252227$ |
$0.477171234$ |
2.992922778 |
\( \frac{2174835996239691808903437496}{5} a^{3} + \frac{5109618269376014117505499997}{5} a^{2} - \frac{1044341186381533012954624973}{5} a - 925687861828768065938355499 \) |
\( \bigl[a^{3} - 3 a\) , \( a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 146 a^{3} - 3 a^{2} - 1259 a - 1300\) , \( 2516 a^{3} - 1161 a^{2} - 24343 a - 22810\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(146a^{3}-3a^{2}-1259a-1300\right){x}+2516a^{3}-1161a^{2}-24343a-22810$ |
| 25.2-d2 |
25.2-d |
$2$ |
$5$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{10} \) |
$17.42076$ |
$(a^3-a^2-3a+1)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1.635450445$ |
$298.2320215$ |
2.992922778 |
\( \frac{943872}{125} a^{3} + \frac{100476501}{3125} a^{2} + \frac{135507253}{3125} a + \frac{12810482}{625} \) |
\( \bigl[a^{3} - 3 a\) , \( a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 6 a^{3} + 7 a^{2} - 24 a - 30\) , \( -9 a^{3} - 6 a^{2} + 52 a + 50\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6a^{3}+7a^{2}-24a-30\right){x}-9a^{3}-6a^{2}+52a+50$ |
| 25.3-a1 |
25.3-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.42076$ |
$(-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$61.37133238$ |
1.882952680 |
\( \frac{8133357}{25} a^{3} + \frac{7223444}{25} a^{2} - \frac{42380619}{25} a - \frac{9156051}{5} \) |
\( \bigl[a^{2} + a - 3\) , \( -a + 1\) , \( a^{2} + a - 3\) , \( -a^{3} + 6 a - 6\) , \( -9 a^{3} + 6 a^{2} + 55 a - 48\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a^{3}+6a-6\right){x}-9a^{3}+6a^{2}+55a-48$ |
| 25.3-b1 |
25.3-b |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$17.42076$ |
$(-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$352.0394050$ |
2.700257252 |
\( -1623 a^{3} + 1898 a^{2} + 4548 a - 114 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( a^{2} - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 3 a - 1\) , \( a^{3} + a^{2} - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(3a-1\right){x}+a^{3}+a^{2}-2$ |
| 25.3-c1 |
25.3-c |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.42076$ |
$(-a^2-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028174239$ |
$2233.566012$ |
1.930744681 |
\( -1623 a^{3} + 1898 a^{2} + 4548 a - 114 \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a + 1\) , \( a^{2} - 3\) , \( 3 a^{3} + 5 a^{2} - 12 a - 15\) , \( a^{3} + 3 a^{2} + a - 2\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a^{3}+5a^{2}-12a-15\right){x}+a^{3}+3a^{2}+a-2$ |
| 25.3-d1 |
25.3-d |
$2$ |
$5$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.42076$ |
$(-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$25$ |
\( 2^{2} \) |
$1$ |
$3.327658221$ |
2.552422574 |
\( \frac{5357496973442499987}{25} a^{3} + \frac{4190018497495156254}{25} a^{2} - \frac{29759807723130703029}{25} a - \frac{6303903372448402516}{5} \) |
\( \bigl[1\) , \( a^{3} - 2 a^{2} - 5 a + 4\) , \( 1\) , \( 1574 a^{3} - 3343 a^{2} - 2624 a + 3703\) , \( 84078 a^{3} - 176952 a^{2} - 135976 a + 202333\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-2a^{2}-5a+4\right){x}^{2}+\left(1574a^{3}-3343a^{2}-2624a+3703\right){x}+84078a^{3}-176952a^{2}-135976a+202333$ |
| 25.3-d2 |
25.3-d |
$2$ |
$5$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{16} \) |
$17.42076$ |
$(-a^2-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$83.19145553$ |
2.552422574 |
\( \frac{3364042267}{9765625} a^{3} + \frac{1964925889}{9765625} a^{2} - \frac{25655280414}{9765625} a - \frac{3815938726}{1953125} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a\) , \( 180 a^{3} + 156 a^{2} - 937 a - 997\) , \( -3246 a^{3} - 2885 a^{2} + 16915 a + 18278\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(180a^{3}+156a^{2}-937a-997\right){x}-3246a^{3}-2885a^{2}+16915a+18278$ |
| 25.4-a1 |
25.4-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.4 |
\( 5^{2} \) |
\( - 5^{11} \) |
$17.42076$ |
$(-a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$158.5324187$ |
2.431990892 |
\( \frac{2539168}{3125} a^{3} - \frac{38097}{625} a^{2} - \frac{12376558}{3125} a - \frac{96508}{3125} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} - a^{2} + 5 a + 5\) , \( a^{2} - a - 4\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-a^{3}-a^{2}+5a+5\right){x}+a^{2}-a-4$ |
| 25.4-b1 |
25.4-b |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.4 |
\( 5^{2} \) |
\( - 5^{9} \) |
$17.42076$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.375589922$ |
$147.7765870$ |
3.405833500 |
\( 8392 a^{3} - 12699 a^{2} - 31219 a + 32013 \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{3} - a^{2} - 3 a + 3\) , \( -2 a^{3} + 3 a - 1\) , \( 2 a^{3} + 6 a^{2} + a - 7\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-2a^{3}+3a-1\right){x}+2a^{3}+6a^{2}+a-7$ |
| 25.4-c1 |
25.4-c |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
25.4 |
\( 5^{2} \) |
\( - 5^{3} \) |
$17.42076$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.035621762$ |
$1850.825470$ |
4.045616628 |
\( 8392 a^{3} - 12699 a^{2} - 31219 a + 32013 \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} + a - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 2 a^{3} + 2 a^{2} - 2 a - 2\) , \( 2 a^{3} + 3 a^{2} - a - 3\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(2a^{3}+2a^{2}-2a-2\right){x}+2a^{3}+3a^{2}-a-3$ |
| 29.1-a1 |
29.1-a |
$2$ |
$7$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
29.1 |
\( 29 \) |
\( - 29^{7} \) |
$17.74698$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 7 \) |
$34.60283518$ |
$0.496286227$ |
3.688211361 |
\( -\frac{832091487795032448748695}{17249876309} a^{3} + \frac{952452924346543210616817}{17249876309} a^{2} + \frac{3902323982359776752395076}{17249876309} a - \frac{3634700568460906339366354}{17249876309} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( a^{3} - 4 a\) , \( 220 a^{3} - 839 a^{2} + 894 a - 288\) , \( 13962 a^{3} - 36614 a^{2} + 1364 a + 17881\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-1\right){x}^{2}+\left(220a^{3}-839a^{2}+894a-288\right){x}+13962a^{3}-36614a^{2}+1364a+17881$ |
| 29.1-a2 |
29.1-a |
$2$ |
$7$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
29.1 |
\( 29 \) |
\( -29 \) |
$17.74698$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 1 \) |
$4.943262169$ |
$1191.583231$ |
3.688211361 |
\( -\frac{5830815}{29} a^{3} + \frac{12208042}{29} a^{2} + \frac{9440576}{29} a - \frac{13931924}{29} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a\) , \( a\) , \( a^{3} - a^{2} - 3 a + 4\) , \( a + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(a^{3}-a^{2}-3a+4\right){x}+a+1$ |
| 35.1-a1 |
35.1-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{2} \cdot 7^{4} \) |
$18.16909$ |
$(-a^2-a+1), (-a^2+2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.012322197$ |
$778.3125948$ |
4.707995898 |
\( -\frac{2689847267}{60025} a^{3} - \frac{4889356739}{60025} a^{2} + \frac{2990070714}{60025} a + \frac{600314061}{12005} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} + 4 a + 2\) , \( a^{2} + a - 2\) , \( -a^{3} + 3 a + 6\) , \( -a^{3} + 2 a\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+4a+2\right){x}^{2}+\left(-a^{3}+3a+6\right){x}-a^{3}+2a$ |
| 35.1-b1 |
35.1-b |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7 \) |
$18.16909$ |
$(-a^2-a+1), (-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$521.1499638$ |
1.998695244 |
\( \frac{27430252697}{175} a^{3} + \frac{63588221624}{175} a^{2} - \frac{13392634399}{175} a - \frac{11500649631}{35} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( -a^{2} + a + 3\) , \( a^{2} + a - 3\) , \( 32 a^{3} + 22 a^{2} - 162 a - 156\) , \( -105 a^{3} - 97 a^{2} + 550 a + 599\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(32a^{3}+22a^{2}-162a-156\right){x}-105a^{3}-97a^{2}+550a+599$ |
| 35.1-b2 |
35.1-b |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{4} \cdot 7^{2} \) |
$18.16909$ |
$(-a^2-a+1), (-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$130.2874909$ |
1.998695244 |
\( -\frac{466157335960529377}{30625} a^{3} + \frac{975698402232038991}{30625} a^{2} + \frac{754743208638389409}{30625} a - \frac{222714784109930694}{6125} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( 0\) , \( -24 a^{3} + 23 a^{2} + 82 a - 89\) , \( 95 a^{3} - 61 a^{2} - 422 a + 341\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(-24a^{3}+23a^{2}+82a-89\right){x}+95a^{3}-61a^{2}-422a+341$ |
| 35.2-a1 |
35.2-a |
$6$ |
$8$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( 5^{4} \cdot 7^{24} \) |
$18.16909$ |
$(-a), (-a^2+2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2^{3} \) |
$1$ |
$1.297843901$ |
3.405400877 |
\( \frac{32728218660976547790359493918362129}{119738269612854009000625} a^{3} - \frac{7492465431863259614064265174522541}{23947653922570801800125} a^{2} - \frac{153488091320022581739677375063611174}{119738269612854009000625} a + \frac{142961792192240963441509624448584601}{119738269612854009000625} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a + 1\) , \( a^{3} - 4 a - 1\) , \( -319 a^{3} + 373 a^{2} + 1472 a - 1480\) , \( -6825 a^{3} + 7663 a^{2} + 31677 a - 29854\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-319a^{3}+373a^{2}+1472a-1480\right){x}-6825a^{3}+7663a^{2}+31677a-29854$ |
| 35.2-a2 |
35.2-a |
$6$ |
$8$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( 5^{16} \cdot 7^{6} \) |
$18.16909$ |
$(-a), (-a^2+2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2^{5} \) |
$1$ |
$1.297843901$ |
3.405400877 |
\( -\frac{3585247840952368237359880243889}{17951812744140625} a^{3} - \frac{636925777301211827706408090419}{3590362548828125} a^{2} + \frac{18682855637509995210492555477734}{17951812744140625} a + \frac{20180930106138994722812261655959}{17951812744140625} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a + 1\) , \( a^{3} - 4 a - 1\) , \( 51 a^{3} - 107 a^{2} - 858 a - 860\) , \( 331 a^{3} - 2973 a^{2} - 13527 a - 12252\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(51a^{3}-107a^{2}-858a-860\right){x}+331a^{3}-2973a^{2}-13527a-12252$ |
| 35.2-a3 |
35.2-a |
$6$ |
$8$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( 5^{8} \cdot 7^{12} \) |
$18.16909$ |
$(-a), (-a^2+2)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
|
\( 2^{4} \) |
$1$ |
$20.76550242$ |
3.405400877 |
\( -\frac{26272660039629428360701}{5406752812890625} a^{3} - \frac{4043491754565015418471}{1081350562578125} a^{2} + \frac{131498317239123577810681}{5406752812890625} a + \frac{144947374979836634419706}{5406752812890625} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a + 1\) , \( a^{3} - 4 a - 1\) , \( -14 a^{3} + 13 a^{2} + 27 a - 130\) , \( -133 a^{3} + 121 a^{2} + 445 a - 819\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-14a^{3}+13a^{2}+27a-130\right){x}-133a^{3}+121a^{2}+445a-819$ |
| 35.2-a4 |
35.2-a |
$6$ |
$8$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( 5^{4} \cdot 7^{6} \) |
$18.16909$ |
$(-a), (-a^2+2)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
|
\( 2^{3} \) |
$1$ |
$332.2480388$ |
3.405400877 |
\( -\frac{97324940346078959821}{73530625} a^{3} + \frac{40741472723672536884}{14706125} a^{2} + \frac{157577019764360193651}{73530625} a - \frac{232493902488955374949}{73530625} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a + 1\) , \( a^{3} - 4 a - 1\) , \( a^{3} - 2 a^{2} - 8 a\) , \( -a^{3} + 4 a^{2} + 5 a - 17\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a^{3}-2a^{2}-8a\right){x}-a^{3}+4a^{2}+5a-17$ |
| 35.2-a5 |
35.2-a |
$6$ |
$8$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{3} \) |
$18.16909$ |
$(-a), (-a^2+2)$ |
$0 \le r \le 1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$2657.984310$ |
3.405400877 |
\( \frac{2973679069}{8575} a^{3} - \frac{1137526636}{1715} a^{2} - \frac{4946267314}{8575} a + \frac{6873052111}{8575} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a + 1\) , \( a^{3} - 4 a - 1\) , \( a^{3} - 2 a^{2} - 8 a + 5\) , \( 2 a^{3} + a^{2} - 7 a + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a^{3}-2a^{2}-8a+5\right){x}+2a^{3}+a^{2}-7a+1$ |
| 35.2-a6 |
35.2-a |
$6$ |
$8$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{3} \) |
$18.16909$ |
$(-a), (-a^2+2)$ |
$0 \le r \le 1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$41.53100485$ |
3.405400877 |
\( -\frac{188705302051003216666447059069}{8575} a^{3} + \frac{78994468365560748894226377241}{1715} a^{2} + \frac{305529281357657504285836775289}{8575} a - \frac{450787146985663246275135964886}{8575} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a + 1\) , \( a^{3} - 4 a - 1\) , \( 16 a^{3} - 17 a^{2} - 43 a + 50\) , \( -a^{3} + 19 a^{2} - 47 a - 7\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(16a^{3}-17a^{2}-43a+50\right){x}-a^{3}+19a^{2}-47a-7$ |
| 49.1-a1 |
49.1-a |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{16} \) |
$18.94956$ |
$(-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1.502002642$ |
$27.14285726$ |
5.003341023 |
\( \frac{500718065734107078}{282475249} a^{3} + \frac{1176400322528301747}{282475249} a^{2} - \frac{240441686104387112}{282475249} a - \frac{1065616788500768418}{282475249} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( 0\) , \( 72 a^{3} - 156 a^{2} - 112 a + 184\) , \( 412 a^{3} - 858 a^{2} - 677 a + 967\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(72a^{3}-156a^{2}-112a+184\right){x}+412a^{3}-858a^{2}-677a+967$ |
| 49.1-a2 |
49.1-a |
$2$ |
$2$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{11} \) |
$18.94956$ |
$(-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.751001321$ |
$217.1428581$ |
5.003341023 |
\( \frac{224443276}{16807} a^{3} + \frac{251959446}{16807} a^{2} - \frac{427552997}{16807} a + \frac{85530617}{16807} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 5 a\) , \( a^{3} - 4 a\) , \( -27 a^{3} - 68 a^{2} + 6 a + 68\) , \( -286 a^{3} - 677 a^{2} + 129 a + 617\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-27a^{3}-68a^{2}+6a+68\right){x}-286a^{3}-677a^{2}+129a+617$ |
| 49.1-b1 |
49.1-b |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$18.94956$ |
$(-a^2+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$654.2274060$ |
5.018137948 |
\( -134309 a^{3} + 314883 a^{2} + 228822 a - 357707 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( a^{3} - 2 a^{2} - 4 a + 4\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 5 a^{3} - 3 a^{2} - 26 a - 5\) , \( 26 a^{3} + 19 a^{2} - 130 a - 121\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+4\right){x}^{2}+\left(5a^{3}-3a^{2}-26a-5\right){x}+26a^{3}+19a^{2}-130a-121$ |
| 49.1-c1 |
49.1-c |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$18.94956$ |
$(-a^2+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.077600834$ |
$755.8589826$ |
1.799620875 |
\( -134309 a^{3} + 314883 a^{2} + 228822 a - 357707 \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - 2\) , \( 3 a^{3} - 3 a^{2} - 16 a + 14\) , \( 3 a^{3} - 4 a^{2} - 14 a + 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(3a^{3}-3a^{2}-16a+14\right){x}+3a^{3}-4a^{2}-14a+13$ |
| 65.1-a1 |
65.1-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.1 |
\( 5 \cdot 13 \) |
\( 5^{2} \cdot 13 \) |
$19.63084$ |
$(-a), (-a^2+a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.424789000$ |
$200.5942464$ |
5.228722389 |
\( \frac{3584239}{325} a^{3} + \frac{748439}{65} a^{2} - \frac{18962934}{325} a - \frac{22787259}{325} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( a^{3} - a^{2} - 3 a + 3\) , \( -a^{3} + 2 a^{2} + 2 a - 4\) , \( a^{2} - a - 6\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-1\right){x}^{2}+\left(-a^{3}+2a^{2}+2a-4\right){x}+a^{2}-a-6$ |
| 65.3-a1 |
65.3-a |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 5^{2} \cdot 13 \) |
$19.63084$ |
$(-a^2-a+1), (-a^2+a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$223.0344024$ |
3.421493469 |
\( \frac{29968}{325} a^{3} - \frac{31819}{325} a^{2} - \frac{56031}{325} a + \frac{10666}{65} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - 4 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 3 a^{2} - 3 a + 11\) , \( 3 a^{3} - 2 a^{2} - 13 a + 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(a^{3}-3a^{2}-3a+11\right){x}+3a^{3}-2a^{2}-13a+2$ |
| 65.3-b1 |
65.3-b |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.3 |
\( 5 \cdot 13 \) |
\( - 5 \cdot 13 \) |
$19.63084$ |
$(-a^2-a+1), (-a^2+a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.104811896$ |
$1686.407225$ |
5.423090965 |
\( \frac{1624107}{65} a^{3} - \frac{2899366}{65} a^{2} - \frac{4411534}{65} a + \frac{1011694}{13} \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 4 a - 1\) , \( -2 a^{3} + 10 a + 1\) , \( a^{3} - 5 a - 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-2a^{3}+10a+1\right){x}+a^{3}-5a-2$ |
| 65.3-c1 |
65.3-c |
$1$ |
$1$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 5 \cdot 13 \) |
$19.63084$ |
$(-a^2-a+1), (-a^2+a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$258.6234969$ |
1.983726716 |
\( -\frac{16838656}{65} a^{3} + \frac{19202048}{65} a^{2} + \frac{78917632}{65} a - \frac{14667776}{13} \) |
\( \bigl[0\) , \( a^{3} - 5 a - 1\) , \( a^{2} - 2\) , \( -a^{3} - 2 a^{2} + 4 a + 8\) , \( a^{3} + a^{2} - 4 a - 5\bigr] \) |
${y}^2+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-a^{3}-2a^{2}+4a+8\right){x}+a^{3}+a^{2}-4a-5$ |
| 65.4-a1 |
65.4-a |
$4$ |
$4$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.4 |
\( 5 \cdot 13 \) |
\( 5^{2} \cdot 13^{6} \) |
$19.63084$ |
$(-a^2-a+1), (a^2-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$487.5193796$ |
2.804574690 |
\( \frac{776612990480795393}{120670225} a^{3} + \frac{1824595606645922956}{120670225} a^{2} - \frac{372924299061361056}{120670225} a - \frac{330554173549087744}{24134045} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 3 a - 1\) , \( 16 a^{3} + 2 a^{2} - 71 a - 59\) , \( -18 a^{3} - 39 a^{2} + 120 a + 161\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(16a^{3}+2a^{2}-71a-59\right){x}-18a^{3}-39a^{2}+120a+161$ |
| 65.4-a2 |
65.4-a |
$4$ |
$4$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.4 |
\( 5 \cdot 13 \) |
\( 5^{4} \cdot 13^{3} \) |
$19.63084$ |
$(-a^2-a+1), (a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$121.8798449$ |
2.804574690 |
\( \frac{1289941579948037059134462661}{1373125} a^{3} + \frac{3030623492863980927681403412}{1373125} a^{2} - \frac{619421014869634331473841487}{1373125} a - \frac{549045199311902353082622508}{274625} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 3 a - 1\) , \( -29 a^{3} - 78 a^{2} + 209 a + 306\) , \( -189 a^{3} + 125 a^{2} + 627 a + 282\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(-29a^{3}-78a^{2}+209a+306\right){x}-189a^{3}+125a^{2}+627a+282$ |
| 65.4-a3 |
65.4-a |
$4$ |
$4$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.4 |
\( 5 \cdot 13 \) |
\( 5 \cdot 13^{12} \) |
$19.63084$ |
$(-a^2-a+1), (a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$121.8798449$ |
2.804574690 |
\( -\frac{356680486461060464501}{116490425612405} a^{3} + \frac{941903782083023180588}{116490425612405} a^{2} - \frac{98580664613082535073}{116490425612405} a - \frac{59471420742679494596}{23298085122481} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{3} + 2 a^{2} + 5 a - 4\) , \( a^{3} - a^{2} - 3 a + 3\) , \( -60 a^{3} + 49 a^{2} + 253 a - 232\) , \( 527 a^{3} - 561 a^{2} - 2415 a + 2225\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-4\right){x}^{2}+\left(-60a^{3}+49a^{2}+253a-232\right){x}+527a^{3}-561a^{2}-2415a+2225$ |
| 65.4-a4 |
65.4-a |
$4$ |
$4$ |
4.4.16997.1 |
$4$ |
$[4, 0]$ |
65.4 |
\( 5 \cdot 13 \) |
\( 5 \cdot 13^{3} \) |
$19.63084$ |
$(-a^2-a+1), (a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$487.5193796$ |
2.804574690 |
\( -\frac{281562743}{10985} a^{3} - \frac{571569841}{10985} a^{2} + \frac{606244686}{10985} a + \frac{238663290}{2197} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{3} + 2 a^{2} + 5 a - 4\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 4 a^{2} + 3 a - 2\) , \( 5 a^{2} + 3 a - 6\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-4\right){x}^{2}+\left(4a^{2}+3a-2\right){x}+5a^{2}+3a-6$ |
*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.
