| 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 |
| 19.2-a1 |
19.2-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( - 19^{3} \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.188035883$ |
$453.8575537$ |
2.299168041 |
\( -\frac{43567442560}{6859} a^{3} + \frac{129020350240}{6859} a^{2} + \frac{163951874560}{6859} a - \frac{485555172640}{6859} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{2} + a - \frac{5}{2}\) , \( \frac{1}{2} a^{2} - \frac{7}{2}\) , \( 3 a^{3} + \frac{11}{2} a^{2} - 11 a - \frac{41}{2}\) , \( 3 a^{3} + 13 a^{2} - 11 a - 49\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{7}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}+a-\frac{5}{2}\right){x}^{2}+\left(3a^{3}+\frac{11}{2}a^{2}-11a-\frac{41}{2}\right){x}+3a^{3}+13a^{2}-11a-49$ |
| 19.2-a2 |
19.2-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( 19^{6} \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.094017941$ |
$453.8575537$ |
2.299168041 |
\( \frac{20257762240}{47045881} a^{3} - \frac{110308292960}{47045881} a^{2} - \frac{75361538240}{47045881} a + \frac{599800342560}{47045881} \) |
\( \bigl[a + 1\) , \( -\frac{1}{2} a^{3} + \frac{7}{2} a - 1\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( -\frac{65}{2} a^{3} - 96 a^{2} + \frac{233}{2} a + 357\) , \( -\frac{1285}{2} a^{3} - 1842 a^{2} + \frac{4841}{2} a + 6931\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{7}{2}a-1\right){x}^{2}+\left(-\frac{65}{2}a^{3}-96a^{2}+\frac{233}{2}a+357\right){x}-\frac{1285}{2}a^{3}-1842a^{2}+\frac{4841}{2}a+6931$ |
| 19.2-a3 |
19.2-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( 19^{2} \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.282053824$ |
$453.8575537$ |
2.299168041 |
\( -\frac{651376965440}{361} a^{3} - \frac{1263931654240}{361} a^{2} + \frac{5364784901440}{361} a + \frac{10409827460640}{361} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a + \frac{9}{2}\) , \( \frac{1}{2} a^{2} - \frac{5}{2}\) , \( -10 a^{3} + \frac{29}{2} a^{2} + 51 a - \frac{57}{2}\) , \( -\frac{7}{2} a^{3} - \frac{17}{2} a^{2} + \frac{61}{2} a + \frac{131}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{9}{2}a+\frac{9}{2}\right){x}^{2}+\left(-10a^{3}+\frac{29}{2}a^{2}+51a-\frac{57}{2}\right){x}-\frac{7}{2}a^{3}-\frac{17}{2}a^{2}+\frac{61}{2}a+\frac{131}{2}$ |
| 19.2-a4 |
19.2-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( -19 \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.564107649$ |
$453.8575537$ |
2.299168041 |
\( \frac{7038892130560}{19} a^{3} - \frac{13656052294240}{19} a^{2} - \frac{57972794066560}{19} a + \frac{112472175052640}{19} \) |
\( \bigl[a + 1\) , \( -\frac{1}{2} a^{2} + a + \frac{5}{2}\) , \( \frac{1}{2} a^{2} + a - \frac{5}{2}\) , \( -\frac{7}{2} a^{3} - \frac{21}{2} a^{2} + \frac{13}{2} a + \frac{51}{2}\) , \( 6 a^{3} + \frac{37}{2} a^{2} - 16 a - \frac{117}{2}\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(\frac{1}{2}a^{2}+a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+a+\frac{5}{2}\right){x}^{2}+\left(-\frac{7}{2}a^{3}-\frac{21}{2}a^{2}+\frac{13}{2}a+\frac{51}{2}\right){x}+6a^{3}+\frac{37}{2}a^{2}-16a-\frac{117}{2}$ |
| 19.2-b1 |
19.2-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( - 19^{3} \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$553.8502367$ |
1.243430488 |
\( -\frac{43567442560}{6859} a^{3} + \frac{129020350240}{6859} a^{2} + \frac{163951874560}{6859} a - \frac{485555172640}{6859} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} + \frac{7}{2} a - 1\) , \( \frac{1}{2} a^{2} - \frac{7}{2}\) , \( -a^{3} - \frac{5}{2} a^{2} + 9 a + \frac{35}{2}\) , \( -\frac{1}{2} a^{3} - \frac{7}{2} a^{2} + \frac{7}{2} a + \frac{61}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{7}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{7}{2}a-1\right){x}^{2}+\left(-a^{3}-\frac{5}{2}a^{2}+9a+\frac{35}{2}\right){x}-\frac{1}{2}a^{3}-\frac{7}{2}a^{2}+\frac{7}{2}a+\frac{61}{2}$ |
| 19.2-b2 |
19.2-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( 19^{6} \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$276.9251183$ |
1.243430488 |
\( \frac{20257762240}{47045881} a^{3} - \frac{110308292960}{47045881} a^{2} - \frac{75361538240}{47045881} a + \frac{599800342560}{47045881} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{9}{2} a + \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( -\frac{173}{2} a^{3} - 253 a^{2} + \frac{637}{2} a + 953\) , \( -\frac{5327}{2} a^{3} - \frac{15295}{2} a^{2} + \frac{20037}{2} a + \frac{57563}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{9}{2}a+\frac{7}{2}\right){x}^{2}+\left(-\frac{173}{2}a^{3}-253a^{2}+\frac{637}{2}a+953\right){x}-\frac{5327}{2}a^{3}-\frac{15295}{2}a^{2}+\frac{20037}{2}a+\frac{57563}{2}$ |
| 19.2-b3 |
19.2-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( 19^{2} \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$276.9251183$ |
1.243430488 |
\( -\frac{651376965440}{361} a^{3} - \frac{1263931654240}{361} a^{2} + \frac{5364784901440}{361} a + \frac{10409827460640}{361} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{9}{2} a - 1\) , \( \frac{1}{2} a^{2} - \frac{5}{2}\) , \( -18 a^{3} + \frac{107}{2} a^{2} + 68 a - \frac{391}{2}\) , \( -16 a^{3} + \frac{101}{2} a^{2} + 59 a - \frac{381}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{5}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{9}{2}a-1\right){x}^{2}+\left(-18a^{3}+\frac{107}{2}a^{2}+68a-\frac{391}{2}\right){x}-16a^{3}+\frac{101}{2}a^{2}+59a-\frac{381}{2}$ |
| 19.2-b4 |
19.2-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.2 |
\( 19 \) |
\( -19 \) |
$14.37784$ |
$(1/2a^2+a-5/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$553.8502367$ |
1.243430488 |
\( \frac{7038892130560}{19} a^{3} - \frac{13656052294240}{19} a^{2} - \frac{57972794066560}{19} a + \frac{112472175052640}{19} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a + \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( -\frac{29}{2} a^{3} - 41 a^{2} + \frac{123}{2} a + 171\) , \( 62 a^{3} + \frac{355}{2} a^{2} - 235 a - \frac{1349}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{9}{2}a+\frac{5}{2}\right){x}^{2}+\left(-\frac{29}{2}a^{3}-41a^{2}+\frac{123}{2}a+171\right){x}+62a^{3}+\frac{355}{2}a^{2}-235a-\frac{1349}{2}$ |
*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.
