| 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 |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{12} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.793500282$ |
1.458435572 |
\( \frac{33777701165536011237}{2213314919066161} a^{2} - \frac{277355301401585077884}{2213314919066161} a + \frac{184960608190583522339}{2213314919066161} \) |
\( \bigl[a^{2} + a - 2\) , \( -a + 1\) , \( a^{2} - 2\) , \( 22 a^{2} - a - 116\) , \( 36 a^{2} - 32 a - 143\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(22a^{2}-a-116\right){x}+36a^{2}-32a-143$ |
| 19.1-a2 |
19.1-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{4} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$23.38050084$ |
1.458435572 |
\( -\frac{28175458029840}{130321} a^{2} + \frac{8070882399369}{130321} a + \frac{118461749726429}{130321} \) |
\( \bigl[a^{2} + a - 2\) , \( -a + 1\) , \( a^{2} - 2\) , \( 22 a^{2} - 6 a - 91\) , \( 79 a^{2} - 21 a - 332\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(22a^{2}-6a-91\right){x}+79a^{2}-21a-332$ |
| 19.1-a3 |
19.1-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$46.76100169$ |
1.458435572 |
\( \frac{2816089}{361} a^{2} - \frac{532982}{361} a - \frac{11269471}{361} \) |
\( \bigl[a^{2} + a - 2\) , \( -a + 1\) , \( a^{2} - 2\) , \( 2 a^{2} - a - 6\) , \( 2 a^{2} - 7\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a^{2}-a-6\right){x}+2a^{2}-7$ |
| 19.1-a4 |
19.1-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$15.58700056$ |
1.458435572 |
\( \frac{108976900762638027}{47045881} a^{2} + \frac{130629686944915561}{47045881} a - \frac{148693269376177306}{47045881} \) |
\( \bigl[a^{2} + a - 2\) , \( -a + 1\) , \( a^{2} - 2\) , \( -8 a^{2} - a + 29\) , \( -3 a^{2} - 9 a - 5\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a^{2}-a+29\right){x}-3a^{2}-9a-5$ |
| 19.1-b1 |
19.1-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{12} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.157201887$ |
0.777957386 |
\( \frac{33777701165536011237}{2213314919066161} a^{2} - \frac{277355301401585077884}{2213314919066161} a + \frac{184960608190583522339}{2213314919066161} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -30 a^{2} + 79 a - 51\) , \( -206 a^{2} + 565 a - 298\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-30a^{2}+79a-51\right){x}-206a^{2}+565a-298$ |
| 19.1-b2 |
19.1-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{4} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$112.2444509$ |
0.777957386 |
\( -\frac{28175458029840}{130321} a^{2} + \frac{8070882399369}{130321} a + \frac{118461749726429}{130321} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 4 a - 11\) , \( a^{2} - 7 a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(4a-11\right){x}+a^{2}-7a+9$ |
| 19.1-b3 |
19.1-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$224.4889018$ |
0.777957386 |
\( \frac{2816089}{361} a^{2} - \frac{532982}{361} a - \frac{11269471}{361} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -a - 1\) , \( -a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}-a$ |
| 19.1-b4 |
19.1-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.34008$ |
$(a^2+a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$8.314403774$ |
0.777957386 |
\( \frac{108976900762638027}{47045881} a^{2} + \frac{130629686944915561}{47045881} a - \frac{148693269376177306}{47045881} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -5 a^{2} - a + 4\) , \( -14 a^{2} + 6 a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-5a^{2}-a+4\right){x}-14a^{2}+6a+3$ |
*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.
