# Properties

 Label 100325.a.100325.1 Conductor 100325 Discriminant 100325 Mordell-Weil group $$\Z$$ 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

 $y^2 + x^2y = x^5 - 2x^4 + 4x^3 - 4x^2 + 3x - 1$ (homogenize, simplify) $y^2 + x^2zy = x^5z - 2x^4z^2 + 4x^3z^3 - 4x^2z^4 + 3xz^5 - z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 7x^4 + 16x^3 - 16x^2 + 12x - 4$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -4, 4, -2, 1], R![0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -4, 4, -2, 1]), R([0, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-4, 12, -16, 16, -7, 4]))

## Invariants

 Conductor: $$N$$ = $$100325$$ = $$5^{2} \cdot 4013$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$100325$$ = $$5^{2} \cdot 4013$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$1664$$ = $$2^{7} \cdot 13$$ $$I_4$$ = $$70720$$ = $$2^{6} \cdot 5 \cdot 13 \cdot 17$$ $$I_6$$ = $$38080960$$ = $$2^{6} \cdot 5 \cdot 59 \cdot 2017$$ $$I_{10}$$ = $$410931200$$ = $$2^{12} \cdot 5^{2} \cdot 4013$$ $$J_2$$ = $$208$$ = $$2^{4} \cdot 13$$ $$J_4$$ = $$1066$$ = $$2 \cdot 13 \cdot 41$$ $$J_6$$ = $$-2719$$ = $$- 2719$$ $$J_8$$ = $$-425477$$ = $$- 13 \cdot 23 \cdot 1423$$ $$J_{10}$$ = $$100325$$ = $$5^{2} \cdot 4013$$ $$g_1$$ = $$389328928768/100325$$ $$g_2$$ = $$9592840192/100325$$ $$g_3$$ = $$-117634816/100325$$

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

All points: $$(1 : 0 : 0),\, (2 : 3 : 1),\, (2 : -7 : 1)$$

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

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 + z^3$$ $$0.396627$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.396627$$ Real period: $$7.575309$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$3.004577$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$2$$ $$2$$ $$1$$ $$1 + 2 T + 5 T^{2}$$
$$4013$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 36 T + 4013 T^{2} )$$

## 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 $$\Q$$.