# Properties

 Label 100010.a.400040.1 Conductor 100010 Discriminant -400040 Mordell-Weil group $$\Z \times \Z/{2}\Z$$ 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

 $y^2 + (x^3 + 1)y = -x^6 + x^5 + 2x^4 + x^3 - 10x^2 + 2x + 5$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -x^6 + x^5z + 2x^4z^2 + x^3z^3 - 10x^2z^4 + 2xz^5 + 5z^6$ (dehomogenize, simplify) $y^2 = -3x^6 + 4x^5 + 8x^4 + 6x^3 - 40x^2 + 8x + 21$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, 2, -10, 1, 2, 1, -1], R![1, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 2, -10, 1, 2, 1, -1]), R([1, 0, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([21, 8, -40, 6, 8, 4, -3]))

## Invariants

 Conductor: $$N$$ = $$100010$$ = $$2 \cdot 5 \cdot 73 \cdot 137$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-400040$$ = $$- 2^{3} \cdot 5 \cdot 73 \cdot 137$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$21736$$ = $$2^{3} \cdot 11 \cdot 13 \cdot 19$$ $$I_4$$ = $$-69788$$ = $$- 2^{2} \cdot 73 \cdot 239$$ $$I_6$$ = $$-552591640$$ = $$- 2^{3} \cdot 5 \cdot 59 \cdot 234149$$ $$I_{10}$$ = $$-1638563840$$ = $$- 2^{15} \cdot 5 \cdot 73 \cdot 137$$ $$J_2$$ = $$2717$$ = $$11 \cdot 13 \cdot 19$$ $$J_4$$ = $$308314$$ = $$2 \cdot 154157$$ $$J_6$$ = $$46839264$$ = $$2^{5} \cdot 3 \cdot 31 \cdot 15739$$ $$J_8$$ = $$8051189423$$ = $$4219 \cdot 1908317$$ $$J_{10}$$ = $$-400040$$ = $$- 2^{3} \cdot 5 \cdot 73 \cdot 137$$ $$g_1$$ = $$-148063561656653357/400040$$ $$g_2$$ = $$-3091947885524641/200020$$ $$g_3$$ = $$-43221451942812/50005$$

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: $$(1 : 0 : 1),\, (1 : -2 : 1),\, (21 : -5666 : 13),\, (21 : -5792 : 13)$$

magma: [C![1,-2,1],C![1,0,1],C![21,-5792,13],C![21,-5666,13]];

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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(1 : 0 : 1) - D_\infty$$ $$(x - z)^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-4xz^2 + 4z^3$$ $$0.656191$$ $$\infty$$
$$D_0 - D_\infty$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$

2-torsion field: splitting field of $$x^{6} - 3 x^{5} - 50 x^{4} + 105 x^{3} + 1525 x^{2} - 1578 x - 19526$$ with Galois group $S_4\times C_2$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.656191$$ Real period: $$5.872743$$ Tamagawa product: $$3$$ Torsion order: $$2$$ Leading coefficient: $$2.890231$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$3$$ $$1$$ $$3$$ $$( 1 - T )( 1 - T + 2 T^{2} )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 5 T^{2} )$$
$$73$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 10 T + 73 T^{2} )$$
$$137$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 10 T + 137 T^{2} )$$

## 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 $$\Q$$.