# Properties

 Label 1575.a.165375.1 Conductor 1575 Discriminant -165375 Mordell-Weil group $$\Z \times \Z/{2}\Z \times \Z/{2}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x + 1)y = -x^5 + 2x^4 + x^2 - 2x$ (homogenize, simplify) $y^2 + (x^2z + xz^2 + z^3)y = -x^5z + 2x^4z^2 + x^2z^4 - 2xz^5$ (dehomogenize, simplify) $y^2 = -4x^5 + 9x^4 + 2x^3 + 7x^2 - 6x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, -6, 7, 2, 9, -4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1575$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 7$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-165375$$ $$=$$ $$- 3^{3} \cdot 5^{3} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-24$$ $$=$$ $$- 2^{3} \cdot 3$$ $$I_4$$ $$=$$ $$77220$$ $$=$$ $$2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \cdot 13$$ $$I_6$$ $$=$$ $$-20329560$$ $$=$$ $$- 2^{3} \cdot 3^{2} \cdot 5 \cdot 149 \cdot 379$$ $$I_{10}$$ $$=$$ $$-677376000$$ $$=$$ $$- 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 7^{2}$$ $$J_2$$ $$=$$ $$-3$$ $$=$$ $$-3$$ $$J_4$$ $$=$$ $$-804$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 67$$ $$J_6$$ $$=$$ $$34624$$ $$=$$ $$2^{6} \cdot 541$$ $$J_8$$ $$=$$ $$-187572$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 \cdot 29$$ $$J_{10}$$ $$=$$ $$-165375$$ $$=$$ $$- 3^{3} \cdot 5^{3} \cdot 7^{2}$$ $$g_1$$ $$=$$ $$9/6125$$ $$g_2$$ $$=$$ $$-804/6125$$ $$g_3$$ $$=$$ $$-34624/18375$$

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
$$(1 : 0 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(2 : 0 : 1)$$ $$(-1 : 2 : 1)$$
$$(-1 : -3 : 1)$$ $$(1 : -3 : 1)$$ $$(2 : -7 : 1)$$ $$(1 : -42 : 4)$$

magma: [C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-42,4],C![1,-3,1],C![1,0,0],C![1,0,1],C![2,-7,1],C![2,0,1]];

Number of rational Weierstrass points: $$2$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) + (1 : -3 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2$$ $$0.027182$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(1 : -42 : 4) - (1 : 0 : 0)$$ $$4x - z$$ $$=$$ $$0,$$ $$32y$$ $$=$$ $$-21z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$3$$ Regulator: $$0.027182$$ Real period: $$17.77977$$ Tamagawa product: $$8$$ Torsion order: $$4$$ Leading coefficient: $$0.241645$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$2$$ $$3$$ $$2$$ $$1 + 2 T + 3 T^{2}$$
$$5$$ $$2$$ $$3$$ $$2$$ $$1 + 2 T + 5 T^{2}$$
$$7$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + 2 T + 7 T^{2} )$$

## 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 $$\Q$$.