# Properties

 Label 12544.g.175616.1 Conductor $12544$ Discriminant $-175616$ Mordell-Weil group $$\Z \oplus \Z/{2}\Z$$ Sato-Tate group $J(E_4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

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Show commands: SageMath / Magma

## Simplified equation

 $y^2 + x^3y = x^5 + x^4 - 2x^2 - 4x - 2$ (homogenize, simplify) $y^2 + x^3y = x^5z + x^4z^2 - 2x^2z^4 - 4xz^5 - 2z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 + 4x^4 - 8x^2 - 16x - 8$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -4, -2, 0, 1, 1]), R([0, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -4, -2, 0, 1, 1], R![0, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-8, -16, -8, 0, 4, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$12544$$ $$=$$ $$2^{8} \cdot 7^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-175616$$ $$=$$ $$- 2^{9} \cdot 7^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$I_4$$ $$=$$ $$-203$$ $$=$$ $$- 7 \cdot 29$$ $$I_6$$ $$=$$ $$455$$ $$=$$ $$5 \cdot 7 \cdot 13$$ $$I_{10}$$ $$=$$ $$686$$ $$=$$ $$2 \cdot 7^{3}$$ $$J_2$$ $$=$$ $$16$$ $$=$$ $$2^{4}$$ $$J_4$$ $$=$$ $$552$$ $$=$$ $$2^{3} \cdot 3 \cdot 23$$ $$J_6$$ $$=$$ $$-5632$$ $$=$$ $$- 2^{9} \cdot 11$$ $$J_8$$ $$=$$ $$-98704$$ $$=$$ $$- 2^{4} \cdot 31 \cdot 199$$ $$J_{10}$$ $$=$$ $$175616$$ $$=$$ $$2^{9} \cdot 7^{3}$$ $$g_1$$ $$=$$ $$2048/343$$ $$g_2$$ $$=$$ $$4416/343$$ $$g_3$$ $$=$$ $$-2816/343$$

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); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_4$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : 1 : 1),\, (-3 : 10 : 1),\, (-3 : 17 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : 1 : 1),\, (-3 : 10 : 1),\, (-3 : 17 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-3 : -7 : 1),\, (-3 : 7 : 1)$$

magma: [C![-3,10,1],C![-3,17,1],C![-1,0,1],C![-1,1,1],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-3,-7,1],C![-3,7,1],C![-1,-1,1],C![-1,1,1],C![1,-1,0],C![1,1,0]]; // simplified model

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.046418$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.046418$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + 2z^3$$ $$0.046418$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 2xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.046418$$ Real period: $$11.29042$$ Tamagawa product: $$6$$ Torsion order: $$2$$ Leading coefficient: $$0.786125$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$8$$ $$9$$ $$3$$ $$1$$
$$7$$ $$2$$ $$3$$ $$2$$ $$1 + 7 T^{2}$$

## Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime $$\ell$$ mod-$$\ell$$ image
$$2$$ 2.90.1
$$3$$ 3.270.1

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $J(E_4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ 4.0.2744.1 with defining polynomial:
$$x^{4} - x^{3} + 3 x^{2} + 3 x + 2$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = -56 b^{3} + 112 b^{2} + 280 b + 112$$
$$g_6 = -812 b^{3} + 8596 b^{2} + 7700 b + 5124$$
Conductor norm: 1024
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = -280 b^{3} + 560 b^{2} - 1288 b + 112$$
$$g_6 = -16338 b^{3} + 25704 b^{2} - 59150 b - 22764$$
Conductor norm: 1024

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ 8.0.481890304.3 with defining polynomial $$x^{8} - 2 x^{7} - 3 x^{6} + 8 x^{5} + x^{4} - 2 x^{3} + 23 x^{2} - 12 x + 2$$

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

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

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a non-Eichler order of index $$4$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

### Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{-2})$$ with generator $$\frac{1}{2} a^{7} - \frac{7}{8} a^{6} - \frac{7}{4} a^{5} + \frac{7}{2} a^{4} + \frac{7}{4} a^{3} - \frac{7}{8} a^{2} + \frac{21}{2} a - \frac{11}{4}$$ with minimal polynomial $$x^{2} + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: E_4
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{14})$$ with generator $$-\frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{5}{4} a^{4} - \frac{5}{4} a^{3} - \frac{5}{2} a^{2} + \frac{3}{2} a - 4$$ with minimal polynomial $$x^{2} - 14$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R$$
Sato Tate group: J(E_2)
Not of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-7})$$ with generator $$\frac{13}{32} a^{7} - \frac{23}{32} a^{6} - \frac{21}{16} a^{5} + \frac{45}{16} a^{4} + \frac{31}{32} a^{3} - \frac{9}{32} a^{2} + \frac{155}{16} a - \frac{33}{16}$$ with minimal polynomial $$x^{2} - x + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R$$
Sato Tate group: J(E_2)
Not of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-2}, \sqrt{-7})$$ with generator $$\frac{3}{32} a^{7} - \frac{5}{32} a^{6} - \frac{7}{16} a^{5} + \frac{11}{16} a^{4} + \frac{25}{32} a^{3} - \frac{19}{32} a^{2} + \frac{13}{16} a + \frac{5}{16}$$ with minimal polynomial $$x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: E_2
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 4.2.21952.1 with generator $$\frac{3}{4} a^{7} - \frac{5}{4} a^{6} - \frac{11}{4} a^{5} + \frac{21}{4} a^{4} + \frac{5}{2} a^{3} - \frac{1}{2} a^{2} + 17 a - 4$$ with minimal polynomial $$x^{4} - 2 x^{3} + 5 x^{2} - 4 x - 10$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{2})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 4.0.2744.1 with generator $$-\frac{3}{16} a^{7} + \frac{1}{4} a^{6} + \frac{7}{8} a^{5} - \frac{5}{4} a^{4} - \frac{21}{16} a^{3} + \frac{1}{4} a^{2} - \frac{29}{8} a + \frac{1}{2}$$ with minimal polynomial $$x^{4} - x^{3} + 3 x^{2} + 3 x + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ 4.0.2744.1 with generator $$\frac{5}{32} a^{7} - \frac{9}{32} a^{6} - \frac{9}{16} a^{5} + \frac{19}{16} a^{4} + \frac{7}{32} a^{3} + \frac{1}{32} a^{2} + \frac{59}{16} a - \frac{23}{16}$$ with minimal polynomial $$x^{4} - x^{3} + 3 x^{2} + 3 x + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ 4.2.21952.1 with generator $$-\frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{5}{16} a^{5} - \frac{1}{16} a^{4} - \frac{35}{32} a^{3} + \frac{9}{32} a^{2} + \frac{1}{16} a - \frac{15}{16}$$ with minimal polynomial $$x^{4} - 2 x^{3} + 5 x^{2} - 4 x - 10$$:

 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{2})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$
Sato Tate group: J(E_1)
Of $$\GL_2$$-type, simple