Properties

 Label 100261.b.100261.1 Conductor 100261 Discriminant -100261 Mordell-Weil group $$\Z$$ 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

 $y^2 + xy = x^5 - x^4 + 3x^2 - 5x + 2$ (homogenize, simplify) $y^2 + xz^2y = x^5z - x^4z^2 + 3x^2z^4 - 5xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 4x^4 + 13x^2 - 20x + 8$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -5, 3, 0, -1, 1], R![0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -5, 3, 0, -1, 1]), R([0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([8, -20, 13, 0, -4, 4]))

Invariants

 Conductor: $$N$$ = $$100261$$ = $$7 \cdot 14323$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-100261$$ = $$- 7 \cdot 14323$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ = $$-2368$$ = $$- 2^{6} \cdot 37$$ $$I_4$$ = $$-9920$$ = $$- 2^{6} \cdot 5 \cdot 31$$ $$I_6$$ = $$-31667264$$ = $$- 2^{6} \cdot 37 \cdot 43 \cdot 311$$ $$I_{10}$$ = $$-410669056$$ = $$- 2^{12} \cdot 7 \cdot 14323$$ $$J_2$$ = $$-296$$ = $$- 2^{3} \cdot 37$$ $$J_4$$ = $$3754$$ = $$2 \cdot 1877$$ $$J_6$$ = $$3441$$ = $$3 \cdot 31 \cdot 37$$ $$J_8$$ = $$-3777763$$ = $$- 11 \cdot 343433$$ $$J_{10}$$ = $$-100261$$ = $$- 7 \cdot 14323$$ $$g_1$$ = $$2272262782976/100261$$ $$g_2$$ = $$97357497344/100261$$ $$g_3$$ = $$-301486656/100261$$

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: $$(1 : 0 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1)$$

magma: [C![1,-1,1],C![1,0,0],C![1,0,1]];

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.095729$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.095729$$ Real period: $$16.88674$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.616554$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - T + 7 T^{2} )$$
$$14323$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 80 T + 14323 T^{2} )$$

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 $$\Q$$.