# Properties

 Label 5911.b.5911.1 Conductor 5911 Discriminant -5911 Mordell-Weil group $$\Z \times \Z$$ 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

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Show commands for: Magma / SageMath

## Simplified equation

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 2, -3, 6, -6, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$5911$$ = $$23 \cdot 257$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-5911$$ = $$- 23 \cdot 257$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-312$$ = $$- 2^{3} \cdot 3 \cdot 13$$ $$I_4$$ = $$-12060$$ = $$- 2^{2} \cdot 3^{2} \cdot 5 \cdot 67$$ $$I_6$$ = $$79848$$ = $$2^{3} \cdot 3^{2} \cdot 1109$$ $$I_{10}$$ = $$-24211456$$ = $$- 2^{12} \cdot 23 \cdot 257$$ $$J_2$$ = $$-39$$ = $$- 3 \cdot 13$$ $$J_4$$ = $$189$$ = $$3^{3} \cdot 7$$ $$J_6$$ = $$1085$$ = $$5 \cdot 7 \cdot 31$$ $$J_8$$ = $$-19509$$ = $$- 3 \cdot 7 \cdot 929$$ $$J_{10}$$ = $$-5911$$ = $$- 23 \cdot 257$$ $$g_1$$ = $$90224199/5911$$ $$g_2$$ = $$11211291/5911$$ $$g_3$$ = $$-1650285/5911$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : -1 : 1)$$ $$(1 : -2 : 1)$$
$$(2 : -4 : 1)$$ $$(2 : -7 : 1)$$ $$(2 : -9 : 3)$$ $$(-3 : 11 : 1)$$ $$(-3 : 18 : 1)$$ $$(2 : -44 : 3)$$

magma: [C![-3,11,1],C![-3,18,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-44,3],C![2,-9,3],C![2,-7,1],C![2,-4,1]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.144667$$ $$\infty$$
$$(0 : 0 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.095839$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.013645$$ Real period: $$19.75200$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.269521$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 23 T^{2} )$$
$$257$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + T + 257 T^{2} )$$

## 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 $$\Q$$.