# Properties

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

# Related objects

Show commands for: SageMath / Magma

## 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) $y^2 = 4x^6 - 24x^4 - 28x^3 - 4x^2 + 4x + 1$ (minimize, homogenize)

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);

### G2 invariants

 $$I_2$$ $$=$$ $$2208$$ $$=$$ $$2^{5} \cdot 3 \cdot 23$$ $$I_4$$ $$=$$ $$200256$$ $$=$$ $$2^{6} \cdot 3 \cdot 7 \cdot 149$$ $$I_6$$ $$=$$ $$116611584$$ $$=$$ $$2^{9} \cdot 3 \cdot 31^{2} \cdot 79$$ $$I_{10}$$ $$=$$ $$-1400897536$$ $$=$$ $$- 2^{23} \cdot 167$$ $$J_2$$ $$=$$ $$276$$ $$=$$ $$2^{2} \cdot 3 \cdot 23$$ $$J_4$$ $$=$$ $$1088$$ $$=$$ $$2^{6} \cdot 17$$ $$J_6$$ $$=$$ $$6144$$ $$=$$ $$2^{11} \cdot 3$$ $$J_8$$ $$=$$ $$128000$$ $$=$$ $$2^{10} \cdot 5^{3}$$ $$J_{10}$$ $$=$$ $$-342016$$ $$=$$ $$- 2^{11} \cdot 167$$ $$g_1$$ $$=$$ $$-1564031349/334$$ $$g_2$$ $$=$$ $$-11169306/167$$ $$g_3$$ $$=$$ $$-228528/167$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$
$$(-1 : -3 : 2)$$ $$(-1 : -5 : 2)$$ $$(-3 : 20 : 1)$$ $$(-3 : -21 : 1)$$

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]];

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
$$(-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.002590$$ $$\infty$$

## 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
$$2$$ $$4$$ $$11$$ $$9$$ $$1$$
$$167$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 8 T + 167 T^{2} )$$

## Sato-Tate group

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

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## 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 $$\Q$$.