# Properties

 Label 26384.d.422144.1 Conductor $26384$ Discriminant $422144$ Mordell-Weil group $$\Z \oplus \Z$$ 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

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$26384$$ $$=$$ $$2^{4} \cdot 17 \cdot 97$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$422144$$ $$=$$ $$2^{8} \cdot 17 \cdot 97$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$24$$ $$=$$ $$2^{3} \cdot 3$$ $$I_4$$ $$=$$ $$180$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5$$ $$I_6$$ $$=$$ $$1224$$ $$=$$ $$2^{3} \cdot 3^{2} \cdot 17$$ $$I_{10}$$ $$=$$ $$-1649$$ $$=$$ $$- 17 \cdot 97$$ $$J_2$$ $$=$$ $$48$$ $$=$$ $$2^{4} \cdot 3$$ $$J_4$$ $$=$$ $$-384$$ $$=$$ $$- 2^{7} \cdot 3$$ $$J_6$$ $$=$$ $$-2048$$ $$=$$ $$- 2^{11}$$ $$J_8$$ $$=$$ $$-61440$$ $$=$$ $$- 2^{12} \cdot 3 \cdot 5$$ $$J_{10}$$ $$=$$ $$-422144$$ $$=$$ $$- 2^{8} \cdot 17 \cdot 97$$ $$g_1$$ $$=$$ $$-995328/1649$$ $$g_2$$ $$=$$ $$165888/1649$$ $$g_3$$ $$=$$ $$18432/1649$$

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

Known points
$$(1 : 0 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$
$$(1 : 1 : 1)$$ $$(3 : -13 : 1)$$ $$(3 : 13 : 1)$$
Known points
$$(1 : 0 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$
$$(1 : 1 : 1)$$ $$(3 : -13 : 1)$$ $$(3 : 13 : 1)$$
Known points
$$(1 : 0 : 0)$$ $$(0 : -1/2 : 1)$$ $$(0 : 1/2 : 1)$$ $$(-1 : -1/2 : 1)$$ $$(-1 : 1/2 : 1)$$ $$(1 : -1/2 : 1)$$
$$(1 : 1/2 : 1)$$ $$(3 : -13/2 : 1)$$ $$(3 : 13/2 : 1)$$

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,0,0],C![1,1,1],C![3,-13,1],C![3,13,1]]; // minimal model

magma: [C![-1,-1/2,1],C![-1,1/2,1],C![0,-1/2,1],C![0,1/2,1],C![1,-1/2,1],C![1,0,0],C![1,1/2,1],C![3,-13/2,1],C![3,13/2,1]]; // 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: $$\Z \oplus \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.194029$$ $$\infty$$
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.088516$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.194029$$ $$\infty$$
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.088516$$ $$\infty$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-1/2xz^2$$ $$0.194029$$ $$\infty$$
$$(1 : -1/2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-1/2z^3$$ $$0.088516$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.017174$$ Real period: $$11.65454$$ Tamagawa product: $$5$$ Torsion order: $$1$$ Leading coefficient: $$1.000796$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$8$$ $$5$$ $$1$$
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 7 T + 17 T^{2} )$$
$$97$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 12 T + 97 T^{2} )$$

## 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?
$$2$$ 2.6.1 no

## 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);