# Properties

 Label 12544.i.614656.1 Conductor $12544$ Discriminant $614656$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{2}\Z$$ Sato-Tate group $E_3$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 = x^5 + 4x^4 - 13x^3 + 9x^2 - x$ (homogenize, simplify) $y^2 = x^5z + 4x^4z^2 - 13x^3z^3 + 9x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = x^5 + 4x^4 - 13x^3 + 9x^2 - x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 9, -13, 4, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 9, -13, 4, 1], R![]);

sage: X = HyperellipticCurve(R([0, -1, 9, -13, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$12544$$ $$=$$ $$2^{8} \cdot 7^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$614656$$ $$=$$ $$2^{8} \cdot 7^{4}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$398$$ $$=$$ $$2 \cdot 199$$ $$I_4$$ $$=$$ $$9016$$ $$=$$ $$2^{3} \cdot 7^{2} \cdot 23$$ $$I_6$$ $$=$$ $$912086$$ $$=$$ $$2 \cdot 7^{2} \cdot 41 \cdot 227$$ $$I_{10}$$ $$=$$ $$2401$$ $$=$$ $$7^{4}$$ $$J_2$$ $$=$$ $$796$$ $$=$$ $$2^{2} \cdot 199$$ $$J_4$$ $$=$$ $$2358$$ $$=$$ $$2 \cdot 3^{2} \cdot 131$$ $$J_6$$ $$=$$ $$-2348$$ $$=$$ $$- 2^{2} \cdot 587$$ $$J_8$$ $$=$$ $$-1857293$$ $$=$$ $$-1857293$$ $$J_{10}$$ $$=$$ $$614656$$ $$=$$ $$2^{8} \cdot 7^{4}$$ $$g_1$$ $$=$$ $$1248318403996/2401$$ $$g_2$$ $$=$$ $$9291226221/4802$$ $$g_3$$ $$=$$ $$-23245787/9604$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_6$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_6$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

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

Number of rational Weierstrass points: $$3$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$19.01761$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$1.188600$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_3$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{7})^+$$ with defining polynomial:
$$x^{3} - x^{2} - 2 x + 1$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = -15349824 b^{2} + 8518608 b + 34491744$$
$$g_6 = 17221522752 b^{2} - 9557224992 b - 38696416416$$
Conductor norm: 4096

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{7})^+$$ with defining polynomial $$x^{3} - x^{2} - 2 x + 1$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$