Properties

 Label 394.a.394.1 Conductor $394$ Discriminant $394$ Mordell-Weil group $$\Z/{10}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

Related objects

Show commands: Magma / SageMath

Simplified equation

 $y^2 + (x^3 + x)y = 2x^5 + x^4 - 12x^3 + 17x - 9$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = 2x^5z + x^4z^2 - 12x^3z^3 + 17xz^5 - 9z^6$ (dehomogenize, simplify) $y^2 = x^6 + 8x^5 + 6x^4 - 48x^3 + x^2 + 68x - 36$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 17, 0, -12, 1, 2]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 17, 0, -12, 1, 2], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-36, 68, 1, -48, 6, 8, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$394$$ $$=$$ $$2 \cdot 197$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$394$$ $$=$$ $$2 \cdot 197$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$11032$$ $$=$$ $$2^{3} \cdot 7 \cdot 197$$ $$I_4$$ $$=$$ $$106300$$ $$=$$ $$2^{2} \cdot 5^{2} \cdot 1063$$ $$I_6$$ $$=$$ $$393913607$$ $$=$$ $$393913607$$ $$I_{10}$$ $$=$$ $$1576$$ $$=$$ $$2^{3} \cdot 197$$ $$J_2$$ $$=$$ $$5516$$ $$=$$ $$2^{2} \cdot 7 \cdot 197$$ $$J_4$$ $$=$$ $$1250044$$ $$=$$ $$2^{2} \cdot 17 \cdot 31 \cdot 593$$ $$J_6$$ $$=$$ $$371875905$$ $$=$$ $$3^{2} \cdot 5 \cdot 8263909$$ $$J_8$$ $$=$$ $$122164372511$$ $$=$$ $$23 \cdot 89 \cdot 127 \cdot 469919$$ $$J_{10}$$ $$=$$ $$394$$ $$=$$ $$2 \cdot 197$$ $$g_1$$ $$=$$ $$12960598758485504$$ $$g_2$$ $$=$$ $$532478222573696$$ $$g_3$$ $$=$$ $$28717744887720$$

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

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

Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ 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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (1 : 0 : 1)$$

magma: [C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model

magma: [C![1,-1,0],C![1,0,1],C![1,1,0]]; // simplified model

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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 8xz - 9z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-11xz^2 + 9z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 8xz - 9z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-11xz^2 + 9z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + 8xz - 9z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 21xz^2 + 18z^3$$ $$0$$ $$10$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$20.07827$$ Tamagawa product: $$1$$ Torsion order: $$10$$ Leading coefficient: $$0.200782$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 2 T^{2} )$$
$$197$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 18 T + 197 T^{2} )$$

Galois representations

The mod-$\ell$ Galois representation has maximal image $$\GSp(4,\F_\ell)$$ for all primes $$\ell$$ except those listed.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.60.1 yes
$$5$$ not computed yes

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

magma: HeuristicDecompositionFactors(C);

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 $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);