| Label |
Class |
Conductor |
Discriminant |
Rank* |
2-Selmer rank |
Torsion |
$\textrm{End}^0(J_{\overline\Q})$ |
$\textrm{End}^0(J)$ |
$\GL_2\textsf{-type}$ |
Sato-Tate |
Nonmaximal primes |
$\Q$-simple |
\(\overline{\Q}\)-simple |
\(\Aut(X)\) |
\(\Aut(X_{\overline{\Q}})\) |
$\Q$-points |
$\Q$-Weierstrass points |
mod-$\ell$ images |
Locally solvable |
Square Ш* |
Analytic Ш* |
Tamagawa |
Regulator |
Real period |
Leading coefficient |
Igusa-Clebsch invariants |
Igusa invariants |
G2-invariants |
Equation |
| 4608.c.27648.1 |
4608.c |
\( 2^{9} \cdot 3^{2} \) |
\( - 2^{10} \cdot 3^{3} \) |
$0$ |
$3$ |
$\Z/2\Z\oplus\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$4$ |
$4$ |
2.360.2, 3.540.5 |
✓ |
✓ |
$1$ |
\( 2^{2} \) |
\(1.000000\) |
\(13.756159\) |
\(0.859760\) |
$[24,-72,-180,108]$ |
$[48,288,-1024,-33024,27648]$ |
$[9216,1152,-\frac{256}{3}]$ |
$y^2 = x^5 - x^4 + x^2 - x$ |
| 4608.c.884736.1 |
4608.c |
\( 2^{9} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{3} \) |
$0$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$C_2$ |
$2$ |
$2$ |
2.180.3, 3.540.5 |
✓ |
✓ |
$1$ |
\( 2 \) |
\(1.000000\) |
\(6.878080\) |
\(0.859760\) |
$[1140,1197,445455,108]$ |
$[4560,853632,210319360,57592172544,884736]$ |
$[2228489100000,91485342000,\frac{14829158000}{3}]$ |
$y^2 = 2x^5 + 7x^4 - 2x^3 - 13x^2 + 10x - 2$ |
| 4608.c.884736.2 |
4608.c |
\( 2^{9} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{3} \) |
$0$ |
$2$ |
$\Z/2\Z\oplus\Z/4\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$C_2$ |
$2$ |
$2$ |
2.180.3, 3.540.5 |
✓ |
✓ |
$1$ |
\( 2^{2} \) |
\(1.000000\) |
\(13.756159\) |
\(0.859760\) |
$[1140,1197,445455,108]$ |
$[4560,853632,210319360,57592172544,884736]$ |
$[2228489100000,91485342000,\frac{14829158000}{3}]$ |
$y^2 = 2x^5 - 7x^4 - 2x^3 + 13x^2 + 10x + 2$ |
| 8192.a.32768.1 |
8192.a |
\( 2^{13} \) |
\( 2^{15} \) |
$0$ |
$3$ |
$\Z/2\Z\oplus\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$4$ |
$4$ |
2.360.2, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2^{2} \) |
\(1.000000\) |
\(16.855414\) |
\(1.053463\) |
$[67,82,1930,4]$ |
$[268,2118,-124,-1129789,32768]$ |
$[\frac{1350125107}{32},\frac{318508017}{256},-\frac{139159}{512}]$ |
$y^2 = x^5 - 3x^3 + 2x$ |
| 8192.a.131072.1 |
8192.a |
\( 2^{13} \) |
\( 2^{17} \) |
$0$ |
$1$ |
$\Z/8\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$C_2$ |
$4$ |
$2$ |
2.180.2, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2^{2} \) |
\(1.000000\) |
\(16.855414\) |
\(1.053463\) |
$[472,7942,1038800,16]$ |
$[1888,63808,910336,-588186624,131072]$ |
$[183020620544,3276205808,24756872]$ |
$y^2 + y = 4x^5 + 15x^4 + 8x^3 - 3x^2 - x$ |
| 12544.g.175616.1 |
12544.g |
\( 2^{8} \cdot 7^{2} \) |
\( - 2^{9} \cdot 7^{3} \) |
$1$ |
$2$ |
$\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$6$ |
$0$ |
2.90.1, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2 \cdot 3 \) |
\(0.046418\) |
\(11.290429\) |
\(0.786126\) |
$[8,-203,455,686]$ |
$[16,552,-5632,-98704,175616]$ |
$[\frac{2048}{343},\frac{4416}{343},-\frac{2816}{343}]$ |
$y^2 + x^3y = x^5 + x^4 - 2x^2 - 4x - 2$ |
| 12800.c.128000.1 |
12800.c |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{3} \) |
$0$ |
$3$ |
$\Z/2\Z\oplus\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$4$ |
$4$ |
2.360.2, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2^{2} \) |
\(1.000000\) |
\(15.952863\) |
\(0.997054\) |
$[104,280,9140,500]$ |
$[208,1056,-1024,-332032,128000]$ |
$[\frac{380204032}{125},\frac{9280128}{125},-\frac{43264}{125}]$ |
$y^2 = x^5 - 3x^4 + 3x^2 - x$ |
| 16384.a.32768.1 |
16384.a |
\( 2^{14} \) |
\( 2^{15} \) |
$0$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$2$ |
$2$ |
2.180.3, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2 \) |
\(1.000000\) |
\(8.427707\) |
\(1.053463\) |
$[67,82,1930,4]$ |
$[268,2118,-124,-1129789,32768]$ |
$[\frac{1350125107}{32},\frac{318508017}{256},-\frac{139159}{512}]$ |
$y^2 = x^5 + 3x^3 + 2x$ |
| 20736.l.373248.1 |
20736.l |
\( 2^{8} \cdot 3^{4} \) |
\( 2^{9} \cdot 3^{6} \) |
$0$ |
$2$ |
$\Z/2\Z\oplus\Z/6\Z$ |
\(\mathsf{QM}\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
✓ |
$C_2$ |
$C_2$ |
$4$ |
$2$ |
2.180.3, 3.480.3 |
✓ |
✓ |
$1$ |
\( 2^{3} \) |
\(1.000000\) |
\(18.600159\) |
\(1.033342\) |
$[146,738,29472,6]$ |
$[876,14262,207364,-5438445,373248]$ |
$[\frac{4146143186}{3},\frac{924693409}{36},\frac{276260689}{648}]$ |
$y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2$ |
| 73728.c.884736.1 |
73728.c |
\( 2^{13} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{3} \) |
$1$ |
$3$ |
$\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$2$ |
$2$ |
2.180.3, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2^{3} \) |
\(0.793842\) |
\(5.583154\) |
\(2.216071\) |
$[195,630,44910,108]$ |
$[780,18630,-380,-86843325,884736]$ |
$[\frac{10442615625}{32},\frac{2558131875}{256},-\frac{401375}{1536}]$ |
$y^2 = x^5 + 5x^3 + 6x$ |
| 73728.d.884736.1 |
73728.d |
\( 2^{13} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{3} \) |
$0$ |
$3$ |
$\Z/2\Z\oplus\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$4$ |
$4$ |
2.360.2, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2^{3} \) |
\(1.000000\) |
\(11.166308\) |
\(1.395789\) |
$[195,630,44910,108]$ |
$[780,18630,-380,-86843325,884736]$ |
$[\frac{10442615625}{32},\frac{2558131875}{256},-\frac{401375}{1536}]$ |
$y^2 = 2x^5 - 5x^3 + 3x$ |
| 147456.c.884736.1 |
147456.c |
\( 2^{14} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{3} \) |
$0$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$2$ |
$2$ |
2.180.3, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2^{2} \) |
\(1.000000\) |
\(5.583154\) |
\(1.395789\) |
$[195,630,44910,108]$ |
$[780,18630,-380,-86843325,884736]$ |
$[\frac{10442615625}{32},\frac{2558131875}{256},-\frac{401375}{1536}]$ |
$y^2 = 2x^5 + 5x^3 + 3x$ |
| 147456.e.884736.1 |
147456.e |
\( 2^{14} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{3} \) |
$1$ |
$3$ |
$\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$4$ |
$2$ |
2.180.3, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2^{2} \) |
\(0.970167\) |
\(11.166308\) |
\(2.708295\) |
$[195,630,44910,108]$ |
$[780,18630,-380,-86843325,884736]$ |
$[\frac{10442615625}{32},\frac{2558131875}{256},-\frac{401375}{1536}]$ |
$y^2 = x^5 - 5x^3 + 6x$ |
| 262144.b.524288.1 |
262144.b |
\( 2^{18} \) |
\( 2^{19} \) |
$1$ |
$2$ |
$\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$2$ |
$2$ |
2.90.3, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2 \) |
\(0.970077\) |
\(7.048011\) |
\(3.418555\) |
$[26,-2,40,2]$ |
$[208,1888,-2304,-1010944,524288]$ |
$[742586,\frac{129623}{4},-\frac{1521}{8}]$ |
$y^2 = x^5 + 2x^3 + 2x$ |
| 262144.c.524288.1 |
262144.c |
\( 2^{18} \) |
\( 2^{19} \) |
$1$ |
$2$ |
$\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$4$ |
$2$ |
2.90.3, 3.270.1 |
✓ |
✓ |
$1$ |
\( 2 \) |
\(0.759196\) |
\(7.048011\) |
\(2.675411\) |
$[26,-2,40,2]$ |
$[208,1888,-2304,-1010944,524288]$ |
$[742586,\frac{129623}{4},-\frac{1521}{8}]$ |
$y^2 = x^5 - 2x^3 + 2x$ |
| 331776.e.995328.1 |
331776.e |
\( 2^{12} \cdot 3^{4} \) |
\( 2^{12} \cdot 3^{5} \) |
$0$ |
$1$ |
$\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$2$ |
$2$ |
2.90.3, 3.540.7 |
✓ |
✓ |
$1$ |
\( 2 \) |
\(1.000000\) |
\(6.346552\) |
\(3.173276\) |
$[58,28,856,16]$ |
$[348,4374,-1836,-4942701,995328]$ |
$[\frac{20511149}{4},\frac{5926527}{32},-\frac{14297}{64}]$ |
$y^2 = x^5 + 3x^3 + 3x$ |
| 331776.g.995328.1 |
331776.g |
\( 2^{12} \cdot 3^{4} \) |
\( 2^{12} \cdot 3^{5} \) |
$1$ |
$2$ |
$\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\Q\) |
|
$J(E_4)$ |
|
✓ |
|
$C_2$ |
$D_4$ |
$4$ |
$2$ |
2.90.3, 3.540.7 |
✓ |
✓ |
$1$ |
\( 2 \) |
\(0.770890\) |
\(6.346552\) |
\(2.446248\) |
$[58,28,856,16]$ |
$[348,4374,-1836,-4942701,995328]$ |
$[\frac{20511149}{4},\frac{5926527}{32},-\frac{14297}{64}]$ |
$y^2 = x^5 - 3x^3 + 3x$ |
