| 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 |
$4$ |
$6$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{2} \) |
$1.77580$ |
$(-a^2-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$148.1410998$ |
0.676506855 |
\( \frac{309847904}{361} a^{2} + \frac{362105728}{361} a - \frac{142642480}{361} \) |
\( \bigl[a^{2} - a - 2\) , \( -1\) , \( a\) , \( -10233 a^{2} - 11974 a + 4716\) , \( 1041051 a^{2} + 1218120 a - 479728\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-10233a^{2}-11974a+4716\right){x}+1041051a^{2}+1218120a-479728$ |
| 19.1-a2 |
19.1-a |
$4$ |
$6$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{6} \) |
$1.77580$ |
$(-a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.486707401$ |
0.676506855 |
\( \frac{15636831814539296}{47045881} a^{2} - \frac{38648046941482112}{47045881} a + \frac{10231591814174352}{47045881} \) |
\( \bigl[a^{2} - a - 2\) , \( -1\) , \( a\) , \( -24578 a^{2} - 28759 a + 11326\) , \( -2418130 a^{2} - 2829421 a + 1114301\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-24578a^{2}-28759a+11326\right){x}-2418130a^{2}-2829421a+1114301$ |
| 19.1-a3 |
19.1-a |
$4$ |
$6$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{3} \) |
$1.77580$ |
$(-a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$10.97341480$ |
0.676506855 |
\( -\frac{72241514230136832}{6859} a^{2} + \frac{49766614320668672}{6859} a + \frac{232207325676879872}{6859} \) |
\( \bigl[0\) , \( -a^{2} + 2\) , \( a^{2} - a - 1\) , \( 20 a^{2} - 9 a - 70\) , \( 81 a^{2} - 52 a - 273\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(20a^{2}-9a-70\right){x}+81a^{2}-52a-273$ |
| 19.1-a4 |
19.1-a |
$4$ |
$6$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$1.77580$ |
$(-a^2-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$296.2821996$ |
0.676506855 |
\( -\frac{16384}{19} a^{2} - \frac{319488}{19} a + \frac{737280}{19} \) |
\( \bigl[0\) , \( -a^{2} + 2\) , \( a^{2} - a - 1\) , \( a\) , \( -1\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+a{x}-1$ |
| 19.1-b1 |
19.1-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{10} \) |
$1.77580$ |
$(-a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \) |
$1.264840851$ |
$2.907033256$ |
0.453363221 |
\( \frac{34600248516351310880}{6131066257801} a^{2} + \frac{14707001073639931296}{6131066257801} a - \frac{7931450838072839760}{6131066257801} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - a - 2\) , \( a^{2} - 2\) , \( -75390 a^{2} - 88214 a + 34739\) , \( -20772669 a^{2} - 24305819 a + 9572276\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-75390a^{2}-88214a+34739\right){x}-20772669a^{2}-24305819a+9572276$ |
| 19.1-b2 |
19.1-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{2} \) |
$1.77580$ |
$(-a^2-a-1)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$0.252968170$ |
$363.3791571$ |
0.453363221 |
\( \frac{19422176}{361} a^{2} - \frac{29688224}{361} a + \frac{7279536}{361} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( -1\) , \( -a^{2} + 3\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}-{x}-a^{2}+3$ |
| 19.1-b3 |
19.1-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$1.77580$ |
$(-a^2-a-1)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$0.505936340$ |
$363.3791571$ |
0.453363221 |
\( -\frac{2985984}{19} a^{2} + \frac{2048000}{19} a + \frac{9617408}{19} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -2423600 a^{2} - 2835822 a + 1116822\) , \( 78861696894 a^{2} + 92275005848 a - 36340347438\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-2423600a^{2}-2835822a+1116822\right){x}+78861696894a^{2}+92275005848a-36340347438$ |
| 19.1-b4 |
19.1-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{5} \) |
$1.77580$ |
$(-a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 1 \) |
$2.529681702$ |
$2.907033256$ |
0.453363221 |
\( \frac{21475865428001583104}{2476099} a^{2} - \frac{53285799291084800000}{2476099} a + \frac{14499029076062806016}{2476099} \) |
\( \bigl[0\) , \( -a^{2} + 2 a + 3\) , \( 1\) , \( -14413186731445 a^{2} - 16864675023680 a + 6641756824108\) , \( -63968762883395881049 a^{2} - 74848985016046516967 a + 29477517729190614798\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-14413186731445a^{2}-16864675023680a+6641756824108\right){x}-63968762883395881049a^{2}-74848985016046516967a+29477517729190614798$ |
*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.
