# Properties

 Label 10044.a.40176.1 Conductor $10044$ Discriminant $-40176$ Mordell-Weil group $$\Z \times \Z/{2}\Z$$ 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

# Related objects

Show commands: SageMath / Magma

## Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -2, 0, 0, -1]), R([1, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -2, 0, 0, -1], R![1, 1, 1]);

sage: X = HyperellipticCurve(R([1, 2, -5, 2, 1, -4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10044$$ $$=$$ $$2^{2} \cdot 3^{4} \cdot 31$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-40176$$ $$=$$ $$- 2^{4} \cdot 3^{4} \cdot 31$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$108$$ $$=$$ $$2^{2} \cdot 3^{3}$$ $$I_4$$ $$=$$ $$-6543$$ $$=$$ $$- 3^{2} \cdot 727$$ $$I_6$$ $$=$$ $$-22041$$ $$=$$ $$- 3^{2} \cdot 31 \cdot 79$$ $$I_{10}$$ $$=$$ $$5142528$$ $$=$$ $$2^{11} \cdot 3^{4} \cdot 31$$ $$J_2$$ $$=$$ $$27$$ $$=$$ $$3^{3}$$ $$J_4$$ $$=$$ $$303$$ $$=$$ $$3 \cdot 101$$ $$J_6$$ $$=$$ $$-1693$$ $$=$$ $$-1693$$ $$J_8$$ $$=$$ $$-34380$$ $$=$$ $$- 2^{2} \cdot 3^{2} \cdot 5 \cdot 191$$ $$J_{10}$$ $$=$$ $$40176$$ $$=$$ $$2^{4} \cdot 3^{4} \cdot 31$$ $$g_1$$ $$=$$ $$177147/496$$ $$g_2$$ $$=$$ $$73629/496$$ $$g_3$$ $$=$$ $$-15237/496$$

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

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

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

Number of rational Weierstrass points: $$1$$

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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.141330$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.141330$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z + xz^2 - z^3$$ $$0.141330$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - xz^2 + z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.141330$$ Real period: $$13.03603$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$0.921192$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$4$$ $$2$$ $$1 + T^{2}$$
$$3$$ $$4$$ $$4$$ $$1$$ $$1 + T$$
$$31$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 8 T + 31 T^{2} )$$

## 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 $$\Q$$.