Properties

 Label 53623.a.53623.1 Conductor 53623 Discriminant -53623 Mordell-Weil group $$\Z \times \Z \times \Z$$ 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

 $y^2 + (x^2 + x + 1)y = x^6 - 2x^4 + x^3 - 2x$ (homogenize, simplify) $y^2 + (x^2z + xz^2 + z^3)y = x^6 - 2x^4z^2 + x^3z^3 - 2xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 - 7x^4 + 6x^3 + 3x^2 - 6x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -6, 3, 6, -7, 0, 4]))

Invariants

 Conductor: $$N$$ = $$53623$$ = $$53623$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-53623$$ = $$- 53623$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ = $$-408$$ = $$- 2^{3} \cdot 3 \cdot 17$$ $$I_4$$ = $$-98076$$ = $$- 2^{2} \cdot 3 \cdot 11 \cdot 743$$ $$I_6$$ = $$-10440600$$ = $$- 2^{3} \cdot 3 \cdot 5^{2} \cdot 17401$$ $$I_{10}$$ = $$-219639808$$ = $$- 2^{12} \cdot 53623$$ $$J_2$$ = $$-51$$ = $$- 3 \cdot 17$$ $$J_4$$ = $$1130$$ = $$2 \cdot 5 \cdot 113$$ $$J_6$$ = $$32292$$ = $$2^{2} \cdot 3^{3} \cdot 13 \cdot 23$$ $$J_8$$ = $$-730948$$ = $$- 2^{2} \cdot 41 \cdot 4457$$ $$J_{10}$$ = $$-53623$$ = $$- 53623$$ $$g_1$$ = $$345025251/53623$$ $$g_2$$ = $$149895630/53623$$ $$g_3$$ = $$-83991492/53623$$

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

Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$
$$(1 : -1 : 1)$$ $$(1 : -2 : 1)$$ $$(-2 : 4 : 1)$$ $$(-2 : -7 : 1)$$ $$(4 : -7 : 3)$$ $$(-1 : 13 : 3)$$
$$(-1 : -34 : 3)$$ $$(4 : -104 : 3)$$ $$(-10 : 945 : 1)$$ $$(-10 : -1036 : 1)$$

magma: [C![-10,-1036,1],C![-10,945,1],C![-2,-7,1],C![-2,4,1],C![-1,-34,3],C![-1,-1,1],C![-1,0,1],C![-1,13,3],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,1,0],C![4,-104,3],C![4,-7,3]];

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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 2z^3$$ $$0.538631$$ $$\infty$$
$$(-1 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.267127$$ $$\infty$$
$$(-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.311205$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.041050$$ Real period: $$16.38656$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.672678$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$53623$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 48 T + 53623 T^{2} )$$

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 $$\Q$$.