# Properties

 Label 2500.a.50000.1 Conductor $2500$ Discriminant $50000$ Mordell-Weil group $$\Z/{15}\Z$$ Sato-Tate group $J(E_1)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

This is a model for the modular curve $X_0(50)$. This is the largest level $N \in \mathbb{N}$ such that $X_0(N)$ has genus 2.

## Simplified equation

 $y^2 + (x^3 + 1)y = x^5 + 2x^3 + x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^5z + 2x^3z^3 + xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 + 10x^3 + 4x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, 4, 0, 10, 0, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$2500$$ $$=$$ $$2^{2} \cdot 5^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$50000$$ $$=$$ $$2^{4} \cdot 5^{5}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$100$$ $$=$$ $$2^{2} \cdot 5^{2}$$ $$I_4$$ $$=$$ $$625$$ $$=$$ $$5^{4}$$ $$I_6$$ $$=$$ $$21385$$ $$=$$ $$5 \cdot 7 \cdot 13 \cdot 47$$ $$I_{10}$$ $$=$$ $$2048$$ $$=$$ $$2^{11}$$ $$J_2$$ $$=$$ $$125$$ $$=$$ $$5^{3}$$ $$J_4$$ $$=$$ $$0$$ $$=$$ $$0$$ $$J_6$$ $$=$$ $$-10000$$ $$=$$ $$- 2^{4} \cdot 5^{4}$$ $$J_8$$ $$=$$ $$-312500$$ $$=$$ $$- 2^{2} \cdot 5^{7}$$ $$J_{10}$$ $$=$$ $$50000$$ $$=$$ $$2^{4} \cdot 5^{5}$$ $$g_1$$ $$=$$ $$9765625/16$$ $$g_2$$ $$=$$ $$0$$ $$g_3$$ $$=$$ $$-3125$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

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: $$\Z/{15}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz + 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2z^3$$ $$0$$ $$15$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$10.23546$$ Tamagawa product: $$15$$ Torsion order: $$15$$ Leading coefficient: $$0.682364$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $J(E_1)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 50.a3
Elliptic curve 50.b3

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{5})$$ with defining polynomial $$x^{2} - x - 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an Eichler order of index $$5$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$