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 |
19.1-a1 |
19.1-a |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{5} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$100$ |
\( 1 \) |
$1$ |
$8.546425211$ |
2.378562937 |
\( -\frac{16075947372441447683071219604597}{2476099} a^{3} - \frac{3746856964444343086453903380667}{2476099} a^{2} + \frac{75759591575353944746601364499055}{2476099} a + \frac{13037311952911337243612323131579}{2476099} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 3\) , \( a^{2} - 3\) , \( -416 a^{3} + 73 a^{2} + 1594 a - 1363\) , \( -4 a^{3} - 6042 a^{2} - 5981 a + 16343\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-416a^{3}+73a^{2}+1594a-1363\right){x}-4a^{3}-6042a^{2}-5981a+16343$ |
19.1-a2 |
19.1-a |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$4$ |
\( 1 \) |
$1$ |
$213.6606302$ |
2.378562937 |
\( \frac{1121092698094}{19} a^{3} - \frac{61710073984}{19} a^{2} - \frac{4164205346487}{19} a + \frac{3126720543036}{19} \) |
\( \bigl[a\) , \( -a^{3} - a^{2} + 4 a + 3\) , \( 0\) , \( 4 a^{3} - 16 a - 8\) , \( 22 a^{3} + 6 a^{2} - 98 a - 29\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}-a^{2}+4a+3\right){x}^{2}+\left(4a^{3}-16a-8\right){x}+22a^{3}+6a^{2}-98a-29$ |
19.1-a3 |
19.1-a |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$427.3212605$ |
2.378562937 |
\( -\frac{4504636}{361} a^{3} + \frac{51759581}{361} a^{2} + \frac{66818656}{361} a - \frac{151977092}{361} \) |
\( \bigl[a\) , \( -a^{3} - a^{2} + 4 a + 3\) , \( 0\) , \( -a^{3} + 4 a + 2\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}-a^{2}+4a+3\right){x}^{2}+\left(-a^{3}+4a+2\right){x}$ |
19.1-a4 |
19.1-a |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{10} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$25$ |
\( 2 \) |
$1$ |
$17.09285042$ |
2.378562937 |
\( -\frac{7453708100651023724731227}{6131066257801} a^{3} - \frac{1737250893512074261087546}{6131066257801} a^{2} + \frac{35126388668209027651898288}{6131066257801} a + \frac{6044827780477046155354082}{6131066257801} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{2} + a - 2\) , \( 659 a^{3} - 621 a^{2} - 3152 a + 3119\) , \( 10968 a^{3} - 11477 a^{2} - 52551 a + 57385\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(659a^{3}-621a^{2}-3152a+3119\right){x}+10968a^{3}-11477a^{2}-52551a+57385$ |
19.1-b1 |
19.1-b |
$2$ |
$2$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$402.9403342$ |
2.242853406 |
\( \frac{921050783480}{361} a^{3} + \frac{1052547388417}{361} a^{2} - \frac{2349627001920}{361} a - \frac{429808776872}{361} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( -a - 1\) , \( a\) , \( -11 a^{2} - 8 a + 29\) , \( -14 a^{3} - 5 a^{2} + 45 a - 22\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11a^{2}-8a+29\right){x}-14a^{3}-5a^{2}+45a-22$ |
19.1-b2 |
19.1-b |
$2$ |
$2$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{4} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$402.9403342$ |
2.242853406 |
\( -\frac{10354365223022612}{130321} a^{3} + \frac{21330013069525904}{130321} a^{2} + \frac{1995387782333409}{130321} a - \frac{338656947464676}{130321} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 32 a^{3} + 19 a^{2} - 155 a - 95\) , \( -218 a^{3} - 25 a^{2} + 1023 a + 41\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(32a^{3}+19a^{2}-155a-95\right){x}-218a^{3}-25a^{2}+1023a+41$ |
19.1-c1 |
19.1-c |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{3} \) |
$11.59825$ |
$(a^2+a-4)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
|
\( 3 \) |
$1$ |
$7.687728166$ |
2.402020858 |
\( \frac{22194249947282956334538362641}{6859} a^{3} - \frac{26005091804226993659513655139}{6859} a^{2} - \frac{106506071026766490646787756502}{6859} a + \frac{129258769902742562322216519975}{6859} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + 5 a\) , \( 1\) , \( 85 a^{3} + 459 a^{2} - 366 a - 2135\) , \( -10862 a^{3} + 5071 a^{2} + 51661 a - 27097\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(85a^{3}+459a^{2}-366a-2135\right){x}-10862a^{3}+5071a^{2}+51661a-27097$ |
19.1-c2 |
19.1-c |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$11.59825$ |
$(a^2+a-4)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
|
\( 2 \cdot 3 \) |
$1$ |
$7.687728166$ |
2.402020858 |
\( -\frac{32920179219067742335}{47045881} a^{3} + \frac{38573217360313423603}{47045881} a^{2} + \frac{157978303983605228544}{47045881} a - \frac{191727709378707559964}{47045881} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + 5 a\) , \( 1\) , \( 40 a^{3} + 39 a^{2} - 181 a - 160\) , \( 261 a^{3} + 179 a^{2} - 1219 a - 762\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(40a^{3}+39a^{2}-181a-160\right){x}+261a^{3}+179a^{2}-1219a-762$ |
19.1-c3 |
19.1-c |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$11.59825$ |
$(a^2+a-4)$ |
$0 \le r \le 1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
|
\( 1 \) |
$1$ |
$622.7059815$ |
2.402020858 |
\( -\frac{140184437194705}{19} a^{3} - \frac{32674219622507}{19} a^{2} + \frac{660633828065334}{19} a + \frac{113692420421896}{19} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{3} + 11 a^{2} + 19 a - 45\) , \( -13 a^{3} + 30 a^{2} + 74 a - 109\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(-a^{3}+11a^{2}+19a-45\right){x}-13a^{3}+30a^{2}+74a-109$ |
19.1-c4 |
19.1-c |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$11.59825$ |
$(a^2+a-4)$ |
$0 \le r \le 1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
|
\( 2 \) |
$1$ |
$622.7059815$ |
2.402020858 |
\( -\frac{468732461}{361} a^{3} - \frac{107902627}{361} a^{2} + \frac{2209071420}{361} a + \frac{374364855}{361} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 4 a^{3} + 6 a^{2} - 11 a - 5\) , \( 7 a^{3} + 9 a^{2} - 18 a - 5\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(4a^{3}+6a^{2}-11a-5\right){x}+7a^{3}+9a^{2}-18a-5$ |
19.1-d1 |
19.1-d |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{3} \) |
$11.59825$ |
$(a^2+a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.422148619$ |
$189.7385750$ |
2.675053088 |
\( \frac{22194249947282956334538362641}{6859} a^{3} - \frac{26005091804226993659513655139}{6859} a^{2} - \frac{106506071026766490646787756502}{6859} a + \frac{129258769902742562322216519975}{6859} \) |
\( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( -406 a^{3} + 495 a^{2} + 1969 a - 2505\) , \( 8935 a^{3} - 10618 a^{2} - 42945 a + 52855\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(-406a^{3}+495a^{2}+1969a-2505\right){x}+8935a^{3}-10618a^{2}-42945a+52855$ |
19.1-d2 |
19.1-d |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$11.59825$ |
$(a^2+a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.211074309$ |
$189.7385750$ |
2.675053088 |
\( -\frac{32920179219067742335}{47045881} a^{3} + \frac{38573217360313423603}{47045881} a^{2} + \frac{157978303983605228544}{47045881} a - \frac{191727709378707559964}{47045881} \) |
\( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( -26 a^{3} + 30 a^{2} + 124 a - 155\) , \( 153 a^{3} - 184 a^{2} - 737 a + 908\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(-26a^{3}+30a^{2}+124a-155\right){x}+153a^{3}-184a^{2}-737a+908$ |
19.1-d3 |
19.1-d |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$11.59825$ |
$(a^2+a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.140716206$ |
$1707.647175$ |
2.675053088 |
\( -\frac{140184437194705}{19} a^{3} - \frac{32674219622507}{19} a^{2} + \frac{660633828065334}{19} a + \frac{113692420421896}{19} \) |
\( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( 14 a^{3} + 10 a^{2} - 66 a - 45\) , \( -101 a^{3} - 46 a^{2} + 475 a + 186\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(14a^{3}+10a^{2}-66a-45\right){x}-101a^{3}-46a^{2}+475a+186$ |
19.1-d4 |
19.1-d |
$4$ |
$6$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$11.59825$ |
$(a^2+a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.070358103$ |
$1707.647175$ |
2.675053088 |
\( -\frac{468732461}{361} a^{3} - \frac{107902627}{361} a^{2} + \frac{2209071420}{361} a + \frac{374364855}{361} \) |
\( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( -a^{3} + 4 a\) , \( -2 a^{3} - a^{2} + 9 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(-a^{3}+4a\right){x}-2a^{3}-a^{2}+9a+4$ |
19.1-e1 |
19.1-e |
$2$ |
$2$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{4} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$99.74404697$ |
0.555197026 |
\( -\frac{10354365223022612}{130321} a^{3} + \frac{21330013069525904}{130321} a^{2} + \frac{1995387782333409}{130321} a - \frac{338656947464676}{130321} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 5 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -16 a^{3} + 12 a^{2} + 72 a - 84\) , \( 11 a^{3} - 32 a^{2} - 66 a + 111\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+5a-1\right){x}^{2}+\left(-16a^{3}+12a^{2}+72a-84\right){x}+11a^{3}-32a^{2}-66a+111$ |
19.1-e2 |
19.1-e |
$2$ |
$2$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$99.74404697$ |
0.555197026 |
\( \frac{921050783480}{361} a^{3} + \frac{1052547388417}{361} a^{2} - \frac{2349627001920}{361} a - \frac{429808776872}{361} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 5 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{3} - 8 a^{2} - 3 a + 21\) , \( -5 a^{3} - 12 a^{2} + 7 a + 20\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+5a-1\right){x}^{2}+\left(-a^{3}-8a^{2}-3a+21\right){x}-5a^{3}-12a^{2}+7a+20$ |
19.1-f1 |
19.1-f |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$2054.093936$ |
0.457341317 |
\( -\frac{4504636}{361} a^{3} + \frac{51759581}{361} a^{2} + \frac{66818656}{361} a - \frac{151977092}{361} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - a - 4\) , \( a^{2} - 3\) , \( 14 a^{3} + 5 a^{2} - 64 a - 10\) , \( -18 a^{3} - 2 a^{2} + 88 a + 12\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(14a^{3}+5a^{2}-64a-10\right){x}-18a^{3}-2a^{2}+88a+12$ |
19.1-f2 |
19.1-f |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$4$ |
\( 1 \) |
$1$ |
$1027.046968$ |
0.457341317 |
\( \frac{1121092698094}{19} a^{3} - \frac{61710073984}{19} a^{2} - \frac{4164205346487}{19} a + \frac{3126720543036}{19} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 2 a^{3} - 18 a^{2} + 24 a - 5\) , \( -14 a^{3} + 48 a^{2} - 41 a + 2\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+1\right){x}^{2}+\left(2a^{3}-18a^{2}+24a-5\right){x}-14a^{3}+48a^{2}-41a+2$ |
19.1-f3 |
19.1-f |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{5} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$100$ |
\( 1 \) |
$1$ |
$1.643275149$ |
0.457341317 |
\( -\frac{16075947372441447683071219604597}{2476099} a^{3} - \frac{3746856964444343086453903380667}{2476099} a^{2} + \frac{75759591575353944746601364499055}{2476099} a + \frac{13037311952911337243612323131579}{2476099} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 3 a - 4\) , \( 0\) , \( -565 a^{3} + 38 a^{2} + 2304 a - 1843\) , \( -5666 a^{3} + 20 a^{2} + 23247 a - 18075\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}+a^{2}+3a-4\right){x}^{2}+\left(-565a^{3}+38a^{2}+2304a-1843\right){x}-5666a^{3}+20a^{2}+23247a-18075$ |
19.1-f4 |
19.1-f |
$4$ |
$10$ |
4.4.8069.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{10} \) |
$11.59825$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$3.286550298$ |
0.457341317 |
\( -\frac{7453708100651023724731227}{6131066257801} a^{3} - \frac{1737250893512074261087546}{6131066257801} a^{2} + \frac{35126388668209027651898288}{6131066257801} a + \frac{6044827780477046155354082}{6131066257801} \) |
\( \bigl[a^{3} - 4 a + 1\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 763 a^{3} - 898 a^{2} - 3634 a + 4396\) , \( -17404 a^{3} + 20349 a^{2} + 83667 a - 101492\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(763a^{3}-898a^{2}-3634a+4396\right){x}-17404a^{3}+20349a^{2}+83667a-101492$ |
*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.