Properties

Label 42048.a.126144.1
Conductor $42048$
Discriminant $126144$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
Sato-Tate group $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = 2x^5 + 3x^4 - 5x^3 - 6x^2 + 8x - 2$ (homogenize, simplify)
$y^2 + x^3y = 2x^5z + 3x^4z^2 - 5x^3z^3 - 6x^2z^4 + 8xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^5 + 12x^4 - 20x^3 - 24x^2 + 32x - 8$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 8, -6, -5, 3, 2]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 8, -6, -5, 3, 2], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-8, 32, -24, -20, 12, 8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(42048\) \(=\) \( 2^{6} \cdot 3^{2} \cdot 73 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(126144\) \(=\) \( 2^{6} \cdot 3^{3} \cdot 73 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2396\) \(=\)  \( 2^{2} \cdot 599 \)
\( I_4 \)  \(=\) \(3436\) \(=\)  \( 2^{2} \cdot 859 \)
\( I_6 \)  \(=\) \(2764040\) \(=\)  \( 2^{3} \cdot 5 \cdot 43 \cdot 1607 \)
\( I_{10} \)  \(=\) \(15768\) \(=\)  \( 2^{3} \cdot 3^{3} \cdot 73 \)
\( J_2 \)  \(=\) \(2396\) \(=\)  \( 2^{2} \cdot 599 \)
\( J_4 \)  \(=\) \(236910\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 53 \cdot 149 \)
\( J_6 \)  \(=\) \(30907908\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 19 \cdot 73 \cdot 619 \)
\( J_8 \)  \(=\) \(4482249867\) \(=\)  \( 3^{2} \cdot 23 \cdot 43 \cdot 103 \cdot 4889 \)
\( J_{10} \)  \(=\) \(126144\) \(=\)  \( 2^{6} \cdot 3^{3} \cdot 73 \)
\( g_1 \)  \(=\) \(1233826502447984/1971\)
\( g_2 \)  \(=\) \(16972374467030/657\)
\( g_3 \)  \(=\) \(4219858561/3\)

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);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

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

Number of rational Weierstrass points: \(0\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.108342\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 + 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.108342\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 + 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.108342\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + 2xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 6xz^2 + 4z^3\) \(0\) \(2\)

2-torsion field: 4.4.10512.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.108342 \)
Real period: \( 12.98322 \)
Tamagawa product: \( 3 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.054976 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(6\) \(1\) \(1 + 2 T + 2 T^{2}\)
\(3\) \(2\) \(3\) \(3\) \(( 1 - T )( 1 + T )\)
\(73\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 14 T + 73 T^{2} )\)

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?
\(2\) 2.90.1 yes
\(3\) 3.45.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-2}) \) with defining polynomial:
  \(x^{2} + 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.0.8.1-657.4-a
  Elliptic curve isogeny class 2.0.8.1-657.3-a

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-2}) \) with defining polynomial \(x^{2} + 2\)

Of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);