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 |
57.1-a1 |
57.1-a |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3^{2} \cdot 19 \) |
$20.85382$ |
$(-1/3a^3-1/3a^2+3a+2), (1/3a^3-2/3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.026666755$ |
$2988.763883$ |
4.528860944 |
\( \frac{764816735}{19} a^{3} - \frac{2749538048}{19} a^{2} + \frac{4046607218}{57} a + \frac{2502325034}{57} \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 4 a\) , \( a^{2} - a - 3\) , \( -\frac{1}{3} a^{3} - \frac{7}{3} a^{2} - 2 a - 4\) , \( 2 a^{3} + 5 a^{2} - 3 a - 3\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+4a\right){x}^{2}+\left(-\frac{1}{3}a^{3}-\frac{7}{3}a^{2}-2a-4\right){x}+2a^{3}+5a^{2}-3a-3$ |
57.1-b1 |
57.1-b |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3^{2} \cdot 19 \) |
$20.85382$ |
$(-1/3a^3-1/3a^2+3a+2), (1/3a^3-2/3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.254490339$ |
$181.8351186$ |
2.629518748 |
\( -\frac{43021}{57} a^{3} + \frac{85696}{57} a^{2} + \frac{369034}{57} a - \frac{454327}{57} \) |
\( \bigl[a + 1\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( a\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( 2 a - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){x}^{2}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}+2a-3$ |
57.1-c1 |
57.1-c |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3^{14} \cdot 19 \) |
$20.85382$ |
$(-1/3a^3-1/3a^2+3a+2), (1/3a^3-2/3a^2-2a+5)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
|
\( 2 \) |
$1$ |
$18.86826300$ |
3.895783893 |
\( -\frac{404963364612649}{4617} a^{3} + \frac{4913008494921869}{41553} a^{2} + \frac{27447645111084058}{41553} a - \frac{31419748638630586}{41553} \) |
\( \bigl[a^{2} - a - 4\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a\) , \( a^{2} - a - 4\) , \( \frac{106}{3} a^{3} + \frac{10}{3} a^{2} - 278 a - 102\) , \( 883 a^{3} + 93 a^{2} - 6961 a - 2401\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+3a\right){x}^{2}+\left(\frac{106}{3}a^{3}+\frac{10}{3}a^{2}-278a-102\right){x}+883a^{3}+93a^{2}-6961a-2401$ |
57.1-d1 |
57.1-d |
$2$ |
$2$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3^{34} \cdot 19^{2} \) |
$20.85382$ |
$(-1/3a^3-1/3a^2+3a+2), (1/3a^3-2/3a^2-2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.949888097$ |
$90.41503854$ |
5.008960080 |
\( \frac{4957694999417395}{46619598843} a^{3} + \frac{9593307021539665}{46619598843} a^{2} - \frac{11878525618289761}{46619598843} a - \frac{5175805466469244}{46619598843} \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( 305 a^{3} + 31 a^{2} - 2403 a - 828\) , \( -3302 a^{3} - 350 a^{2} + 26036 a + 8964\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}^{2}+\left(305a^{3}+31a^{2}-2403a-828\right){x}-3302a^{3}-350a^{2}+26036a+8964$ |
57.1-d2 |
57.1-d |
$2$ |
$2$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{17} \cdot 19 \) |
$20.85382$ |
$(-1/3a^3-1/3a^2+3a+2), (1/3a^3-2/3a^2-2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$3.899776194$ |
$180.8300770$ |
5.008960080 |
\( -\frac{217217028070}{373977} a^{3} + \frac{1324842416596}{373977} a^{2} - \frac{2515509051217}{373977} a + \frac{161738260330}{41553} \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( -\frac{190}{3} a^{3} - \frac{22}{3} a^{2} + 502 a + 172\) , \( -\frac{1208}{3} a^{3} - \frac{134}{3} a^{2} + 3179 a + 1095\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}^{2}+\left(-\frac{190}{3}a^{3}-\frac{22}{3}a^{2}+502a+172\right){x}-\frac{1208}{3}a^{3}-\frac{134}{3}a^{2}+3179a+1095$ |
57.1-e1 |
57.1-e |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3^{6} \cdot 19 \) |
$20.85382$ |
$(-1/3a^3-1/3a^2+3a+2), (1/3a^3-2/3a^2-2a+5)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.199016654$ |
$115.2853003$ |
5.214954858 |
\( \frac{732466}{171} a^{3} + \frac{3616163}{513} a^{2} - \frac{6735095}{513} a - \frac{2042806}{513} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\) , \( a^{2} - 2 a - 5\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 6\) , \( \frac{1}{3} a^{3} + \frac{4}{3} a^{2} - 4 a - 5\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){y}={x}^{3}+\left(a^{2}-2a-5\right){x}^{2}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+6\right){x}+\frac{1}{3}a^{3}+\frac{4}{3}a^{2}-4a-5$ |
57.1-f1 |
57.1-f |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( - 3^{2} \cdot 19 \) |
$20.85382$ |
$(-1/3a^3-1/3a^2+3a+2), (1/3a^3-2/3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.241538489$ |
$306.1728257$ |
4.202233631 |
\( \frac{14227698499096}{57} a^{3} + \frac{9089735445467}{19} a^{2} - \frac{34285899745478}{57} a - \frac{14634193939504}{57} \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 6 a + 3\) , \( a^{2} - a - 3\) , \( 2 a^{3} - 5 a^{2} - 13 a + 16\) , \( \frac{4}{3} a^{3} - \frac{14}{3} a^{2} - 12 a + 22\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(\frac{2}{3}a^{3}-\frac{1}{3}a^{2}-6a+3\right){x}^{2}+\left(2a^{3}-5a^{2}-13a+16\right){x}+\frac{4}{3}a^{3}-\frac{14}{3}a^{2}-12a+22$ |
*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.