# Properties

 Label 18080.c.723200.1 Conductor 18080 Discriminant -723200 Mordell-Weil group $$\Z \times \Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

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Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + xy = x^6 - 2x^5 + x^3 + x^2 - 2x + 1$ (homogenize, simplify) $y^2 + xz^2y = x^6 - 2x^5z + x^3z^3 + x^2z^4 - 2xz^5 + z^6$ (dehomogenize, simplify) $y^2 = 4x^6 - 8x^5 + 4x^3 + 5x^2 - 8x + 4$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([4, -8, 5, 4, 0, -8, 4]))

## Invariants

 Conductor: $$N$$ = $$18080$$ = $$2^{5} \cdot 5 \cdot 113$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-723200$$ = $$- 2^{8} \cdot 5^{2} \cdot 113$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-1184$$ = $$- 2^{5} \cdot 37$$ $$I_4$$ = $$186688$$ = $$2^{6} \cdot 2917$$ $$I_6$$ = $$-76176896$$ = $$- 2^{9} \cdot 148783$$ $$I_{10}$$ = $$-2962227200$$ = $$- 2^{20} \cdot 5^{2} \cdot 113$$ $$J_2$$ = $$-148$$ = $$- 2^{2} \cdot 37$$ $$J_4$$ = $$-1032$$ = $$- 2^{3} \cdot 3 \cdot 43$$ $$J_6$$ = $$44800$$ = $$2^{8} \cdot 5^{2} \cdot 7$$ $$J_8$$ = $$-1923856$$ = $$- 2^{4} \cdot 11 \cdot 17 \cdot 643$$ $$J_{10}$$ = $$-723200$$ = $$- 2^{8} \cdot 5^{2} \cdot 113$$ $$g_1$$ = $$277375828/2825$$ $$g_2$$ = $$-13068474/2825$$ $$g_3$$ = $$-153328/113$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## 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

Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : 0 : 1)$$ $$(1 : -1 : 1)$$
$$(-1 : -2 : 1)$$ $$(-1 : 3 : 1)$$ $$(1 : 3 : 2)$$ $$(3 : -1 : 2)$$ $$(1 : -7 : 2)$$ $$(3 : -11 : 2)$$
$$(7 : -41 : 6)$$ $$(7 : -211 : 6)$$

magma: [C![-1,-2,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-7,2],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![1,3,2],C![3,-11,2],C![3,-1,2],C![7,-211,6],C![7,-41,6]];

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.557085$$ $$\infty$$
$$(0 : -1 : 1) + (3 : -11 : 2) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(-2x + 3z) x$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$-xz^2 - 4z^3$$ $$0.039896$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 - 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$3$$ Regulator: $$0.022196$$ Real period: $$14.76743$$ Tamagawa product: $$8$$ Torsion order: $$2$$ Leading coefficient: $$0.655575$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$8$$ $$5$$ $$4$$ $$1 + T$$
$$5$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$113$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 14 T + 113 T^{2} )$$

## 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 $$\Q$$.