# Properties

 Label 100162.b.400648.1 Conductor 100162 Discriminant -400648 Mordell-Weil group $$\Z \times \Z/{3}\Z$$ 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![2, 5, 4, 0, -4, 1], R![0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 5, 4, 0, -4, 1]), R([0, 1, 1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 5, 4, 0, -4, 1], R![0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 20, 17, 2, -15, 4]))

 $y^2 + (x^2 + x)y = x^5 - 4x^4 + 4x^2 + 5x + 2$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z - 4x^4z^2 + 4x^2z^4 + 5xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 15x^4 + 2x^3 + 17x^2 + 20x + 8$ (minimize, homogenize)

## Invariants

 $$N$$ = $$100162$$ = $$2 \cdot 61 \cdot 821$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-400648$$ = $$- 2^{3} \cdot 61 \cdot 821$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$7304$$ = $$2^{3} \cdot 11 \cdot 83$$ $$I_4$$ = $$-928124$$ = $$- 2^{2} \cdot 331 \cdot 701$$ $$I_6$$ = $$-2399465848$$ = $$- 2^{3} \cdot 13 \cdot 23071787$$ $$I_{10}$$ = $$-1641054208$$ = $$- 2^{15} \cdot 61 \cdot 821$$ $$J_2$$ = $$913$$ = $$11 \cdot 83$$ $$J_4$$ = $$44400$$ = $$2^{4} \cdot 3 \cdot 5^{2} \cdot 37$$ $$J_6$$ = $$3475524$$ = $$2^{2} \cdot 3 \cdot 13 \cdot 22279$$ $$J_8$$ = $$300448353$$ = $$3 \cdot 43 \cdot 293 \cdot 7949$$ $$J_{10}$$ = $$-400648$$ = $$- 2^{3} \cdot 61 \cdot 821$$ $$g_1$$ = $$-634386434595793/400648$$ $$g_2$$ = $$-4223819158350/50081$$ $$g_3$$ = $$-724272266289/100162$$

## 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![1,-4,1],C![1,0,0],C![1,2,1]];

Points: $$(1 : 0 : 0),\, (1 : 2 : 1),\, (1 : -4 : 1)$$

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

Number of rational Weierstrass points: $$1$$

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: $$\Z \times \Z/{3}\Z$$

Generator Height Order
$$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-4z^3$$ $$0.772491$$ $$\infty$$
$$x^2 - 2xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$3$$

## BSD invariants

 Analytic rank: $$1$$   (upper bound) Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.772491$$ Real period: $$11.20513$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$2.885288$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$3$$ $$1$$ $$3$$ $$( 1 - T )( 1 + T + 2 T^{2} )$$
$$61$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 13 T + 61 T^{2} )$$
$$821$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 21 T + 821 T^{2} )$$

## 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 $$\Q$$.