Properties

Label 4608.c.27648.1
Conductor $4608$
Discriminant $-27648$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Sato-Tate group $J(E_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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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)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 0, -1, 1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 0, -1, 1], R![]);
 
sage: X = HyperellipticCurve(R([0, -1, 1, 0, -1, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

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

\( I_2 \)  \(=\) \(24\) \(=\)  \( 2^{3} \cdot 3 \)
\( I_4 \)  \(=\) \(-72\) \(=\)  \( - 2^{3} \cdot 3^{2} \)
\( I_6 \)  \(=\) \(-180\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 5 \)
\( I_{10} \)  \(=\) \(108\) \(=\)  \( 2^{2} \cdot 3^{3} \)
\( J_2 \)  \(=\) \(48\) \(=\)  \( 2^{4} \cdot 3 \)
\( J_4 \)  \(=\) \(288\) \(=\)  \( 2^{5} \cdot 3^{2} \)
\( J_6 \)  \(=\) \(-1024\) \(=\)  \( - 2^{10} \)
\( J_8 \)  \(=\) \(-33024\) \(=\)  \( - 2^{8} \cdot 3 \cdot 43 \)
\( J_{10} \)  \(=\) \(27648\) \(=\)  \( 2^{10} \cdot 3^{3} \)
\( g_1 \)  \(=\) \(9216\)
\( g_2 \)  \(=\) \(1152\)
\( g_3 \)  \(=\) \(-256/3\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

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

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (1 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (1 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (1 : 0 : 1)\)

magma: [C![-1,0,1],C![0,0,1],C![1,0,0],C![1,0,1]]; // minimal model
 
magma: [C![-1,0,1],C![0,0,1],C![1,0,0],C![1,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(4\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

magma: HeuristicDecompositionFactors(C);
 

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 \(x^{2} + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: E_4
  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 \(x^{2} - 3\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: J(E_2)
  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 \(x^{2} - x + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: J(E_2)
  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 \(x^{4} - x^{2} + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: E_2
  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 \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  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 \(x^{4} - 2 x^{3} - 2 x + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  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 \(x^{4} - 2 x^{3} - 2 x + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  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 \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  Of \(\GL_2\)-type, not simple

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);