Minimal equation
Minimal equation
Simplified equation
$y^2 = x^5 - x^4 + x^2 - x$ | (homogenize, simplify) |
$y^2 = x^5z - x^4z^2 + x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^5 - x^4 + x^2 - x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(4608\) | \(=\) | \( 2^{9} \cdot 3^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-27648\) | \(=\) | \( - 2^{10} \cdot 3^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(24\) | \(=\) | \( 2^{3} \cdot 3 \) |
\( I_4 \) | \(=\) | \(-72\) | \(=\) | \( - 2^{3} \cdot 3^{2} \) |
\( I_6 \) | \(=\) | \(-180\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 5 \) |
\( I_{10} \) | \(=\) | \(108\) | \(=\) | \( 2^{2} \cdot 3^{3} \) |
\( J_2 \) | \(=\) | \(48\) | \(=\) | \( 2^{4} \cdot 3 \) |
\( J_4 \) | \(=\) | \(288\) | \(=\) | \( 2^{5} \cdot 3^{2} \) |
\( J_6 \) | \(=\) | \(-1024\) | \(=\) | \( - 2^{10} \) |
\( J_8 \) | \(=\) | \(-33024\) | \(=\) | \( - 2^{8} \cdot 3 \cdot 43 \) |
\( J_{10} \) | \(=\) | \(27648\) | \(=\) | \( 2^{10} \cdot 3^{3} \) |
\( g_1 \) | \(=\) | \(9216\) | ||
\( g_2 \) | \(=\) | \(1152\) | ||
\( g_3 \) | \(=\) | \(-256/3\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(4\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{-3}) \)
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 13.75615 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.859759 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(9\) | \(10\) | \(2\) | \(1\) | |
\(3\) | \(2\) | \(3\) | \(2\) | \(1 + 3 T^{2}\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.360.2 | yes |
\(3\) | 3.540.5 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(E_4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 4.0.432.1 with defining polynomial:
\(x^{4} - 3 x^{2} + 3\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -480 b^{3} + 672 b^{2} + 576 b - 1008\)
\(g_6 = -16704 b^{3} + 19008 b^{2} + 29952 b - 39744\)
Conductor norm: 1024
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 480 b^{3} + 672 b^{2} - 576 b - 1008\)
\(g_6 = 16704 b^{3} + 19008 b^{2} - 29952 b - 39744\)
Conductor norm: 1024
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.2985984.1 with defining polynomial \(x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} - 4 x + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(\frac{16}{11} a^{7} - 2 a^{6} + \frac{21}{11} a^{5} - \frac{23}{11} a^{4} + 9 a^{3} - \frac{105}{11} a^{2} + \frac{83}{11} a - \frac{30}{11}\) with minimal polynomial \(x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{3}) \) with generator \(2 a^{7} - 4 a^{6} + 3 a^{5} - 3 a^{4} + 13 a^{3} - 19 a^{2} + 11 a - 4\) with minimal polynomial \(x^{2} - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(\frac{18}{11} a^{7} - 2 a^{6} + \frac{14}{11} a^{5} - \frac{19}{11} a^{4} + 10 a^{3} - \frac{92}{11} a^{2} + \frac{48}{11} a - \frac{9}{11}\) with minimal polynomial \(x^{2} - x + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\zeta_{12})\) with generator \(\frac{3}{11} a^{7} - a^{6} + \frac{6}{11} a^{5} - \frac{5}{11} a^{4} + 2 a^{3} - \frac{52}{11} a^{2} + \frac{19}{11} a - \frac{7}{11}\) with minimal polynomial \(x^{4} - x^{2} + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.0.432.1 with generator \(\frac{9}{11} a^{7} - 2 a^{6} + \frac{18}{11} a^{5} - \frac{15}{11} a^{4} + 6 a^{3} - \frac{112}{11} a^{2} + \frac{57}{11} a - \frac{21}{11}\) with minimal polynomial \(x^{4} - 3 x^{2} + 3\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 4.2.1728.1 with generator \(-\frac{8}{11} a^{7} + a^{6} - \frac{5}{11} a^{5} + \frac{6}{11} a^{4} - 4 a^{3} + \frac{47}{11} a^{2} - \frac{14}{11} a + \frac{4}{11}\) with minimal polynomial \(x^{4} - 2 x^{3} - 2 x + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.2.1728.1 with generator \(-a^{7} + a^{6} - a^{5} + a^{4} - 6 a^{3} + 4 a^{2} - 3 a + 1\) with minimal polynomial \(x^{4} - 2 x^{3} - 2 x + 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.0.432.1 with generator \(\frac{9}{11} a^{7} - a^{6} + \frac{7}{11} a^{5} - \frac{15}{11} a^{4} + 5 a^{3} - \frac{46}{11} a^{2} + \frac{24}{11} a - \frac{21}{11}\) with minimal polynomial \(x^{4} - 3 x^{2} + 3\):
\(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple