# Properties

 Label 10025.a.10025.1 Conductor $10025$ Discriminant $10025$ Mordell-Weil group $$\Z$$ 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 for: SageMath / Magma

## Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10025$$ $$=$$ $$5^{2} \cdot 401$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$10025$$ $$=$$ $$5^{2} \cdot 401$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$336$$ $$=$$ $$2^{4} \cdot 3 \cdot 7$$ $$I_4$$ $$=$$ $$1620$$ $$=$$ $$2^{2} \cdot 3^{4} \cdot 5$$ $$I_6$$ $$=$$ $$219240$$ $$=$$ $$2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 29$$ $$I_{10}$$ $$=$$ $$-40100$$ $$=$$ $$- 2^{2} \cdot 5^{2} \cdot 401$$ $$J_2$$ $$=$$ $$168$$ $$=$$ $$2^{3} \cdot 3 \cdot 7$$ $$J_4$$ $$=$$ $$906$$ $$=$$ $$2 \cdot 3 \cdot 151$$ $$J_6$$ $$=$$ $$-784$$ $$=$$ $$- 2^{4} \cdot 7^{2}$$ $$J_8$$ $$=$$ $$-238137$$ $$=$$ $$- 3 \cdot 79379$$ $$J_{10}$$ $$=$$ $$-10025$$ $$=$$ $$- 5^{2} \cdot 401$$ $$g_1$$ $$=$$ $$-133827821568/10025$$ $$g_2$$ $$=$$ $$-4295918592/10025$$ $$g_3$$ $$=$$ $$22127616/10025$$

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

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

magma: [C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,2,0]]; // 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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0.041216$$ $$\infty$$
Generator $D_0$ Height Order
$$2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0.041216$$ $$\infty$$
Generator $D_0$ Height Order
$$2 \cdot(0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 + z^3$$ $$0.041216$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.041216$$ Real period: $$22.57242$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.930347$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$2$$ $$2$$ $$1$$ $$1 + 4 T + 5 T^{2}$$
$$401$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 30 T + 401 T^{2} )$$

## 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 $$\Q$$.