| 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 |
$6$ |
$8$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$6.09043$ |
$(1/2a^3-3/2a^2-1/2a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$217.4274603$ |
1.152363234 |
\( \frac{1598792626}{19} a^{3} - \frac{2783527522}{19} a^{2} - \frac{5527860549}{19} a + \frac{7542480177}{19} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{3}{2} a - 3\) , \( -\frac{1}{2} a^{3} + \frac{3}{2} a^{2} + \frac{3}{2} a - 2\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3\) , \( -3 a^{3} - 3 a^{2} + 5 a + 4\) , \( -16 a^{3} - 23 a^{2} + 25 a + 27\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{3}{2}a-3\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-3\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{3}{2}a^{2}+\frac{3}{2}a-2\right){x}^{2}+\left(-3a^{3}-3a^{2}+5a+4\right){x}-16a^{3}-23a^{2}+25a+27$ |
| 19.1-a2 |
19.1-a |
$6$ |
$8$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{8} \) |
$6.09043$ |
$(1/2a^3-3/2a^2-1/2a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$108.7137301$ |
1.152363234 |
\( \frac{92552153363284660}{16983563041} a^{3} + \frac{66917950344298997}{16983563041} a^{2} - \frac{179110212086206138}{16983563041} a - \frac{19423552243094385}{16983563041} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a\) , \( -a^{2} + a + 3\) , \( a^{2} - 2\) , \( 6 a^{3} - 28 a^{2} + 35 a - 9\) , \( -\frac{139}{2} a^{3} + \frac{313}{2} a^{2} + \frac{267}{2} a - 251\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(6a^{3}-28a^{2}+35a-9\right){x}-\frac{139}{2}a^{3}+\frac{313}{2}a^{2}+\frac{267}{2}a-251$ |
| 19.1-a3 |
19.1-a |
$6$ |
$8$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{4} \) |
$6.09043$ |
$(1/2a^3-3/2a^2-1/2a+4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$434.8549206$ |
1.152363234 |
\( -\frac{6593184883}{130321} a^{3} + \frac{2678496927}{130321} a^{2} + \frac{26352219634}{130321} a + \frac{16630098173}{130321} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a\) , \( -a^{2} + a + 3\) , \( a^{2} - 2\) , \( \frac{17}{2} a^{3} - \frac{51}{2} a^{2} + \frac{5}{2} a + 21\) , \( \frac{67}{2} a^{3} - \frac{189}{2} a^{2} - \frac{3}{2} a + 76\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(\frac{17}{2}a^{3}-\frac{51}{2}a^{2}+\frac{5}{2}a+21\right){x}+\frac{67}{2}a^{3}-\frac{189}{2}a^{2}-\frac{3}{2}a+76$ |
| 19.1-a4 |
19.1-a |
$6$ |
$8$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$6.09043$ |
$(1/2a^3-3/2a^2-1/2a+4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$434.8549206$ |
1.152363234 |
\( -\frac{379239815884}{361} a^{3} + \frac{1045413444949}{361} a^{2} + \frac{59826090630}{361} a - \frac{863574260399}{361} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{7}{2} a + 4\) , \( a + 1\) , \( -a^{3} - a^{2} + 6 a + 5\) , \( \frac{3}{2} a^{3} - \frac{1}{2} a^{2} - \frac{15}{2} a - 5\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{7}{2}a+4\right){x}^{2}+\left(-a^{3}-a^{2}+6a+5\right){x}+\frac{3}{2}a^{3}-\frac{1}{2}a^{2}-\frac{15}{2}a-5$ |
| 19.1-a5 |
19.1-a |
$6$ |
$8$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{2} \) |
$6.09043$ |
$(1/2a^3-3/2a^2-1/2a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$108.7137301$ |
1.152363234 |
\( -\frac{4678351774560}{361} a^{3} - \frac{648252464829}{361} a^{2} + \frac{22653681959498}{361} a + \frac{16435969522901}{361} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + 1\) , \( a^{2} - 2 a - 2\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a\) , \( 2 a^{2} - 2 a - 3\) , \( \frac{1}{2} a^{3} + \frac{3}{2} a^{2} - \frac{3}{2} a - 2\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a+1\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(2a^{2}-2a-3\right){x}+\frac{1}{2}a^{3}+\frac{3}{2}a^{2}-\frac{3}{2}a-2$ |
| 19.1-a6 |
19.1-a |
$6$ |
$8$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$6.09043$ |
$(1/2a^3-3/2a^2-1/2a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$54.35686508$ |
1.152363234 |
\( -\frac{238451843945748300466}{19} a^{3} + \frac{657315936650561061710}{19} a^{2} + \frac{37619193173970978989}{19} a - \frac{542985496932508946577}{19} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a\) , \( a\) , \( 4 a^{3} + a^{2} - 14 a - 8\) , \( -5 a^{3} - 7 a^{2} + 17 a + 23\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a\right){x}^{2}+\left(4a^{3}+a^{2}-14a-8\right){x}-5a^{3}-7a^{2}+17a+23$ |
*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.
