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Properties

Label 2672.a.342016.1
Conductor $2672$
Discriminant $-342016$
Mordell-Weil group \(\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^6 - 6x^4 - 7x^3 - x^2 + x$ (homogenize, simplify)
$y^2 + z^3y = x^6 - 6x^4z^2 - 7x^3z^3 - x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 24x^4 - 28x^3 - 4x^2 + 4x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -1, -7, -6, 0, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -1, -7, -6, 0, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, 4, -4, -28, -24, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2672\) \(=\) \( 2^{4} \cdot 167 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-342016\) \(=\) \( - 2^{11} \cdot 167 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(276\) \(=\)  \( 2^{2} \cdot 3 \cdot 23 \)
\( I_4 \)  \(=\) \(3129\) \(=\)  \( 3 \cdot 7 \cdot 149 \)
\( I_6 \)  \(=\) \(227757\) \(=\)  \( 3 \cdot 31^{2} \cdot 79 \)
\( I_{10} \)  \(=\) \(-42752\) \(=\)  \( - 2^{8} \cdot 167 \)
\( J_2 \)  \(=\) \(276\) \(=\)  \( 2^{2} \cdot 3 \cdot 23 \)
\( J_4 \)  \(=\) \(1088\) \(=\)  \( 2^{6} \cdot 17 \)
\( J_6 \)  \(=\) \(6144\) \(=\)  \( 2^{11} \cdot 3 \)
\( J_8 \)  \(=\) \(128000\) \(=\)  \( 2^{10} \cdot 5^{3} \)
\( J_{10} \)  \(=\) \(-342016\) \(=\)  \( - 2^{11} \cdot 167 \)
\( g_1 \)  \(=\) \(-1564031349/334\)
\( g_2 \)  \(=\) \(-11169306/167\)
\( g_3 \)  \(=\) \(-228528/167\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Rational points

All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\)
\((-1 : -3 : 2)\) \((-1 : -5 : 2)\) \((-3 : 20 : 1)\) \((-3 : -21 : 1)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\)
\((-1 : -3 : 2)\) \((-1 : -5 : 2)\) \((-3 : 20 : 1)\) \((-3 : -21 : 1)\)
All points
\((1 : -2 : 0)\) \((1 : 2 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-1 : -2 : 2)\) \((-1 : 2 : 2)\) \((-3 : -41 : 1)\) \((-3 : 41 : 1)\)

magma: [C![-3,-21,1],C![-3,20,1],C![-1,-5,2],C![-1,-3,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0]]; // minimal model
 
magma: [C![-3,-41,1],C![-3,41,1],C![-1,-2,2],C![-1,2,2],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,2,0]]; // 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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.002590\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.002590\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) + (0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.002590\) \(\infty\)

2-torsion field: 6.4.342016.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.002590 \)
Real period: \( 18.07464 \)
Tamagawa product: \( 9 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.421449 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(11\) \(9\) \(1\)
\(167\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 167 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.10.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

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);