Properties

 Label 1104.a.17664.1 Conductor $1104$ Discriminant $17664$ 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: SageMath / Magma

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -4, 4, -2, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -4, 4, -2, 1], R![]);

sage: X = HyperellipticCurve(R([-1, 3, -4, 4, -2, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$1104$$ $$=$$ $$2^{4} \cdot 3 \cdot 23$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$17664$$ $$=$$ $$2^{8} \cdot 3 \cdot 23$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$88$$ $$=$$ $$2^{3} \cdot 11$$ $$I_4$$ $$=$$ $$160$$ $$=$$ $$2^{5} \cdot 5$$ $$I_6$$ $$=$$ $$4888$$ $$=$$ $$2^{3} \cdot 13 \cdot 47$$ $$I_{10}$$ $$=$$ $$69$$ $$=$$ $$3 \cdot 23$$ $$J_2$$ $$=$$ $$176$$ $$=$$ $$2^{4} \cdot 11$$ $$J_4$$ $$=$$ $$864$$ $$=$$ $$2^{5} \cdot 3^{3}$$ $$J_6$$ $$=$$ $$-1280$$ $$=$$ $$- 2^{8} \cdot 5$$ $$J_8$$ $$=$$ $$-242944$$ $$=$$ $$- 2^{8} \cdot 13 \cdot 73$$ $$J_{10}$$ $$=$$ $$17664$$ $$=$$ $$2^{8} \cdot 3 \cdot 23$$ $$g_1$$ $$=$$ $$659664896/69$$ $$g_2$$ $$=$$ $$6133248/23$$ $$g_3$$ $$=$$ $$-154880/69$$

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: $$(1 : 0 : 0),\, (1 : -1 : 1),\, (1 : 1 : 1)$$ All points: $$(1 : 0 : 0),\, (1 : -1 : 1),\, (1 : 1 : 1)$$ All points: $$(1 : 0 : 0),\, (1 : -1/2 : 1),\, (1 : 1/2 : 1)$$

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

magma: [C![1,-1/2,1],C![1,0,0],C![1,1/2,1]]; // 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
$$(1 : 1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(1 : 1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(1 : 1/2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$1/2z^3$$ $$0$$ $$10$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$8.907497$$ Tamagawa product: $$5$$ Torsion order: $$10$$ Leading coefficient: $$0.445374$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$8$$ $$5$$ $$1$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + T + 3 T^{2} )$$
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 4 T + 23 T^{2} )$$

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 $$\Q$$.