| 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 |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{3} \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.188035883$ |
$453.8575537$ |
2.299168041 |
\( \frac{43567442560}{6859} a^{3} + \frac{129020350240}{6859} a^{2} - \frac{163951874560}{6859} a - \frac{485555172640}{6859} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{7}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( -\frac{1}{2} a^{2} - a + \frac{9}{2}\) , \( \frac{1}{2} a^{2} - \frac{11}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{7}{2}a-\frac{5}{2}\right){x}^{2}+\left(-\frac{1}{2}a^{2}-a+\frac{9}{2}\right){x}+\frac{1}{2}a^{2}-\frac{11}{2}$ |
| 19.1-a2 |
19.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.094017941$ |
$453.8575537$ |
2.299168041 |
\( -\frac{20257762240}{47045881} a^{3} - \frac{110308292960}{47045881} a^{2} + \frac{75361538240}{47045881} a + \frac{599800342560}{47045881} \) |
\( \bigl[a + 1\) , \( \frac{1}{2} a^{3} - \frac{9}{2} a - 1\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( 32 a^{3} - 96 a^{2} - 114 a + 357\) , \( 642 a^{3} - 1842 a^{2} - 2417 a + 6931\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{9}{2}a-1\right){x}^{2}+\left(32a^{3}-96a^{2}-114a+357\right){x}+642a^{3}-1842a^{2}-2417a+6931$ |
| 19.1-a3 |
19.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.282053824$ |
$453.8575537$ |
2.299168041 |
\( \frac{651376965440}{361} a^{3} - \frac{1263931654240}{361} a^{2} - \frac{5364784901440}{361} a + \frac{10409827460640}{361} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a + \frac{9}{2}\) , \( \frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( \frac{7}{2} a^{3} + \frac{27}{2} a^{2} + \frac{5}{2} a - \frac{45}{2}\) , \( 3 a^{3} + 11 a^{2} + 8 a + 2\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{9}{2}a+\frac{9}{2}\right){x}^{2}+\left(\frac{7}{2}a^{3}+\frac{27}{2}a^{2}+\frac{5}{2}a-\frac{45}{2}\right){x}+3a^{3}+11a^{2}+8a+2$ |
| 19.1-a4 |
19.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.564107649$ |
$453.8575537$ |
2.299168041 |
\( -\frac{7038892130560}{19} a^{3} - \frac{13656052294240}{19} a^{2} + \frac{57972794066560}{19} a + \frac{112472175052640}{19} \) |
\( \bigl[a + 1\) , \( -\frac{1}{2} a^{2} + a + \frac{5}{2}\) , \( \frac{1}{2} a^{2} - \frac{7}{2}\) , \( \frac{5}{2} a^{3} - 10 a^{2} + \frac{3}{2} a + 26\) , \( -\frac{33}{2} a^{3} + 52 a^{2} + \frac{83}{2} a - 153\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{7}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+a+\frac{5}{2}\right){x}^{2}+\left(\frac{5}{2}a^{3}-10a^{2}+\frac{3}{2}a+26\right){x}-\frac{33}{2}a^{3}+52a^{2}+\frac{83}{2}a-153$ |
| 19.1-b1 |
19.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{3} \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$553.8502367$ |
1.243430488 |
\( \frac{43567442560}{6859} a^{3} + \frac{129020350240}{6859} a^{2} - \frac{163951874560}{6859} a - \frac{485555172640}{6859} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a - 1\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( -\frac{1}{2} a^{3} - a^{2} + \frac{5}{2} a + 6\) , \( \frac{1}{2} a^{2} + a - \frac{7}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a-1\right){x}^{2}+\left(-\frac{1}{2}a^{3}-a^{2}+\frac{5}{2}a+6\right){x}+\frac{1}{2}a^{2}+a-\frac{7}{2}$ |
| 19.1-b2 |
19.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$276.9251183$ |
1.243430488 |
\( -\frac{20257762240}{47045881} a^{3} - \frac{110308292960}{47045881} a^{2} + \frac{75361538240}{47045881} a + \frac{599800342560}{47045881} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{9}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( 10 a^{3} - \frac{53}{2} a^{2} - 19 a + \frac{139}{2}\) , \( 142 a^{3} - 404 a^{2} - 492 a + 1433\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{9}{2}\right){x}^{2}+\left(10a^{3}-\frac{53}{2}a^{2}-19a+\frac{139}{2}\right){x}+142a^{3}-404a^{2}-492a+1433$ |
| 19.1-b3 |
19.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$276.9251183$ |
1.243430488 |
\( \frac{651376965440}{361} a^{3} - \frac{1263931654240}{361} a^{2} - \frac{5364784901440}{361} a + \frac{10409827460640}{361} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{9}{2} a - 1\) , \( \frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( \frac{37}{2} a^{3} + 52 a^{2} - \frac{137}{2} a - 189\) , \( 50 a^{3} + \frac{289}{2} a^{2} - 188 a - \frac{1097}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{9}{2}a-1\right){x}^{2}+\left(\frac{37}{2}a^{3}+52a^{2}-\frac{137}{2}a-189\right){x}+50a^{3}+\frac{289}{2}a^{2}-188a-\frac{1097}{2}$ |
| 19.1-b4 |
19.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$14.37784$ |
$(1/2a^3-7/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$553.8502367$ |
1.243430488 |
\( -\frac{7038892130560}{19} a^{3} - \frac{13656052294240}{19} a^{2} + \frac{57972794066560}{19} a + \frac{112472175052640}{19} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{9}{2} a + \frac{9}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a\) , \( -a^{3} + 5 a^{2} + 20 a - 53\) , \( -\frac{59}{2} a^{3} + 64 a^{2} + \frac{483}{2} a - 496\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{9}{2}a+\frac{9}{2}\right){x}^{2}+\left(-a^{3}+5a^{2}+20a-53\right){x}-\frac{59}{2}a^{3}+64a^{2}+\frac{483}{2}a-496$ |
*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.
