# Properties

 Label 12544.h.401408.1 Conductor $12544$ Discriminant $401408$ Mordell-Weil group $$\Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: SageMath / Magma

## Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -1, 4, -1, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -1, 4, -1, 1], R![]);

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$12544$$ $$=$$ $$2^{8} \cdot 7^{2}$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(12544,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$401408$$ $$=$$ $$2^{13} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$120$$ $$=$$ $$2^{3} \cdot 3 \cdot 5$$ $$I_4$$ $$=$$ $$408$$ $$=$$ $$2^{3} \cdot 3 \cdot 17$$ $$I_6$$ $$=$$ $$14988$$ $$=$$ $$2^{2} \cdot 3 \cdot 1249$$ $$I_{10}$$ $$=$$ $$1568$$ $$=$$ $$2^{5} \cdot 7^{2}$$ $$J_2$$ $$=$$ $$240$$ $$=$$ $$2^{4} \cdot 3 \cdot 5$$ $$J_4$$ $$=$$ $$1312$$ $$=$$ $$2^{5} \cdot 41$$ $$J_6$$ $$=$$ $$-2048$$ $$=$$ $$- 2^{11}$$ $$J_8$$ $$=$$ $$-553216$$ $$=$$ $$- 2^{8} \cdot 2161$$ $$J_{10}$$ $$=$$ $$401408$$ $$=$$ $$2^{13} \cdot 7^{2}$$ $$g_1$$ $$=$$ $$97200000/49$$ $$g_2$$ $$=$$ $$2214000/49$$ $$g_3$$ $$=$$ $$-14400/49$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

magma: [C![0,0,1],C![1,-2,1],C![1,0,0],C![1,2,1]]; // minimal model

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

Number of rational Weierstrass points: $$2$$

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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.119959$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.119959$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.119959$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.119959$$ Real period: $$7.745608$$ Tamagawa product: $$4$$ Torsion order: $$2$$ Leading coefficient: $$0.929162$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$8$$ $$13$$ $$4$$ $$1$$
$$7$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 112.a
Elliptic curve isogeny class 112.b

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.