# Properties

 Label 10125.a.10125.2 Conductor $10125$ Discriminant $10125$ Mordell-Weil group $$\Z \times \Z/{5}\Z$$ 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

 $y^2 + (x^3 + x^2 + 1)y = 9x^4 + 28x^3 - x^2 - 34x + 13$ (homogenize, simplify) $y^2 + (x^3 + x^2z + z^3)y = 9x^4z^2 + 28x^3z^3 - x^2z^4 - 34xz^5 + 13z^6$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + 37x^4 + 114x^3 - 2x^2 - 136x + 53$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([13, -34, -1, 28, 9]), R([1, 0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![13, -34, -1, 28, 9], R![1, 0, 1, 1]);

sage: X = HyperellipticCurve(R([53, -136, -2, 114, 37, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10125$$ $$=$$ $$3^{4} \cdot 5^{3}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$10125$$ $$=$$ $$3^{4} \cdot 5^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$27780$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 463$$ $$I_4$$ $$=$$ $$151768305$$ $$=$$ $$3^{2} \cdot 5 \cdot 13 \cdot 61 \cdot 4253$$ $$I_6$$ $$=$$ $$775034217225$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 3444596521$$ $$I_{10}$$ $$=$$ $$1296000$$ $$=$$ $$2^{7} \cdot 3^{4} \cdot 5^{3}$$ $$J_2$$ $$=$$ $$6945$$ $$=$$ $$3 \cdot 5 \cdot 463$$ $$J_4$$ $$=$$ $$-4313970$$ $$=$$ $$- 2 \cdot 3^{2} \cdot 5 \cdot 47933$$ $$J_6$$ $$=$$ $$2210480200$$ $$=$$ $$2^{3} \cdot 5^{2} \cdot 1249 \cdot 8849$$ $$J_8$$ $$=$$ $$-814638042975$$ $$=$$ $$- 3 \cdot 5^{2} \cdot 10861840573$$ $$J_{10}$$ $$=$$ $$10125$$ $$=$$ $$3^{4} \cdot 5^{3}$$ $$g_1$$ $$=$$ $$1595755016890725$$ $$g_2$$ $$=$$ $$-142724601457530$$ $$g_3$$ $$=$$ $$94771685998760/9$$

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

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

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

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: $$\Z \times \Z/{5}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$2.350829$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0$$ $$5$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$2.350829$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0$$ $$5$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + x^2z + z^3$$ $$2.350829$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + x^2z + 2xz^2 - z^3$$ $$0$$ $$5$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$2.350829$$ Real period: $$11.74426$$ Tamagawa product: $$1$$ Torsion order: $$5$$ Leading coefficient: $$1.104351$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$4$$ $$4$$ $$1$$ $$1 + T + 3 T^{2}$$
$$5$$ $$3$$ $$3$$ $$1$$ $$1 + T$$

## 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 $$\Q$$.