# Properties

 Label 1145.a.1145.1 Conductor $1145$ Discriminant $1145$ 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

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Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + 1)y = -3x^4 + 3x^3 - x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -3x^4z^2 + 3x^3z^3 - xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 12x^4 + 14x^3 - 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 3, -3]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 3, -3], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([1, -4, 0, 14, -12, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1145$$ $$=$$ $$5 \cdot 229$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$1145$$ $$=$$ $$5 \cdot 229$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$468$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 13$$ $$I_4$$ $$=$$ $$5337$$ $$=$$ $$3^{2} \cdot 593$$ $$I_6$$ $$=$$ $$771165$$ $$=$$ $$3^{2} \cdot 5 \cdot 17137$$ $$I_{10}$$ $$=$$ $$146560$$ $$=$$ $$2^{7} \cdot 5 \cdot 229$$ $$J_2$$ $$=$$ $$117$$ $$=$$ $$3^{2} \cdot 13$$ $$J_4$$ $$=$$ $$348$$ $$=$$ $$2^{2} \cdot 3 \cdot 29$$ $$J_6$$ $$=$$ $$224$$ $$=$$ $$2^{5} \cdot 7$$ $$J_8$$ $$=$$ $$-23724$$ $$=$$ $$- 2^{2} \cdot 3^{2} \cdot 659$$ $$J_{10}$$ $$=$$ $$1145$$ $$=$$ $$5 \cdot 229$$ $$g_1$$ $$=$$ $$21924480357/1145$$ $$g_2$$ $$=$$ $$557361324/1145$$ $$g_3$$ $$=$$ $$3066336/1145$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (1 : -4 : 2),\, (1 : -5 : 2)$$ All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (1 : -4 : 2),\, (1 : -5 : 2)$$ All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (1 : -1 : 2),\, (1 : 1 : 2)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-4,2],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model

magma: [C![0,-1,1],C![0,1,1],C![1,-1,2],C![1,1,2],C![1,-1,0],C![1,0,1],C![1,1,0]]; // 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
$$(1 : -4 : 2) - (1 : -1 : 0)$$ $$z (2x - z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0.026503$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-4xz^2 + z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(1 : -4 : 2) - (1 : -1 : 0)$$ $$z (2x - z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-z^3$$ $$0.026503$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-4xz^2 + z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$z (2x - z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 - z^3$$ $$0.026503$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - 3xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 8xz^2 + 3z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.026503$$ Real period: $$29.58514$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.196027$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 5 T^{2} )$$
$$229$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 229 T^{2} )$$

## 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 $$\Q$$.