# Properties

 Label 925.a.23125.1 Conductor 925 Discriminant 23125 Mordell-Weil group $$\Z/{2}\Z \times \Z/{8}\Z$$ 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

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -5, 18, -19, 1, 5], R![0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -5, 18, -19, 1, 5]), R([0, 1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -5, 18, -19, 1, 5], R![0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -20, 73, -76, 4, 20]))

 $y^2 + xy = 5x^5 + x^4 - 19x^3 + 18x^2 - 5x$ (homogenize, simplify) $y^2 + xz^2y = 5x^5z + x^4z^2 - 19x^3z^3 + 18x^2z^4 - 5xz^5$ (dehomogenize, simplify) $y^2 = 20x^5 + 4x^4 - 76x^3 + 73x^2 - 20x$ (minimize, homogenize)

## Invariants

 $$N$$ = $$925$$ = $$5^{2} \cdot 37$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$23125$$ = $$5^{4} \cdot 37$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$13984$$ = $$2^{5} \cdot 19 \cdot 23$$ $$I_4$$ = $$808576$$ = $$2^{7} \cdot 6317$$ $$I_6$$ = $$3568957120$$ = $$2^{6} \cdot 5 \cdot 11152991$$ $$I_{10}$$ = $$94720000$$ = $$2^{12} \cdot 5^{4} \cdot 37$$ $$J_2$$ = $$1748$$ = $$2^{2} \cdot 19 \cdot 23$$ $$J_4$$ = $$118890$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 1321$$ $$J_6$$ = $$10257041$$ = $$10257041$$ $$J_8$$ = $$948618892$$ = $$2^{2} \cdot 13 \cdot 18242671$$ $$J_{10}$$ = $$23125$$ = $$5^{4} \cdot 37$$ $$g_1$$ = $$16319511005139968/23125$$ $$g_2$$ = $$126998797147776/4625$$ $$g_3$$ = $$31340429803664/23125$$

## Automorphism group

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

## Rational points

magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![4,-50,5]];

Points: $$(0 : 0 : 1),\, (1 : 0 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (4 : -50 : 5)$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$3$$

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);

This curve is locally solvable everywhere.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{2}\Z \times \Z/{8}\Z$$

Generator Height Order
$$(-5x + 4z) x$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
$$(x - z) (5x - 4z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 - 2z^3$$ $$0$$ $$8$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$20.87893$$ Tamagawa product: $$4$$ Torsion order: $$16$$ Leading coefficient: $$0.326233$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$4$$ $$2$$ $$4$$ $$( 1 - T )( 1 + T )$$
$$37$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 6 T + 37 T^{2} )$$

## 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 $$\Q$$.