# Properties

 Label 448.a.448.2 Conductor $448$ Discriminant $-448$ Mordell-Weil group $$\Z/{12}\Z$$ Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{CM} \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + x)y = -2x^4 + 7$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -2x^4z^2 + 7z^6$ (dehomogenize, simplify) $y^2 = x^6 - 6x^4 + x^2 + 28$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([7, 0, 0, 0, -2]), R([0, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![7, 0, 0, 0, -2], R![0, 1, 0, 1]);

sage: X = HyperellipticCurve(R([28, 0, 1, 0, -6, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$448$$ $$=$$ $$2^{6} \cdot 7$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-448$$ $$=$$ $$- 2^{6} \cdot 7$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$828$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 23$$ $$I_4$$ $$=$$ $$16635$$ $$=$$ $$3 \cdot 5 \cdot 1109$$ $$I_6$$ $$=$$ $$5308452$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 147457$$ $$I_{10}$$ $$=$$ $$56$$ $$=$$ $$2^{3} \cdot 7$$ $$J_2$$ $$=$$ $$828$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 23$$ $$J_4$$ $$=$$ $$17476$$ $$=$$ $$2^{2} \cdot 17 \cdot 257$$ $$J_6$$ $$=$$ $$-853888$$ $$=$$ $$- 2^{7} \cdot 7 \cdot 953$$ $$J_8$$ $$=$$ $$-253107460$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 43 \cdot 294311$$ $$J_{10}$$ $$=$$ $$448$$ $$=$$ $$2^{6} \cdot 7$$ $$g_1$$ $$=$$ $$6080953884912/7$$ $$g_2$$ $$=$$ $$155007628668/7$$ $$g_3$$ $$=$$ $$-1306723104$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : 0 : 1),\, (2 : 0 : 1)$$

magma: [C![-2,5,1],C![1,-1,0],C![1,0,0],C![2,-5,1]]; // minimal model

magma: [C![-2,0,1],C![1,-1,0],C![1,1,0],C![2,0,1]]; // 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: $$\Z/{12}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(2 : -5 : 1) - (1 : 0 : 0)$$ $$z (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + 3z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$(2 : -5 : 1) - (1 : 0 : 0)$$ $$z (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + 3z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$(2 : 0 : 1) - (1 : 1 : 0)$$ $$z (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + xz^2 + 6z^3$$ $$0$$ $$12$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$31.17115$$ Tamagawa product: $$1$$ Torsion order: $$12$$ Leading coefficient: $$0.216466$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$6$$ $$6$$ $$1$$ $$1 + T$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 7 T^{2} )$$

## 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
$$2$$ 2.90.3
$$3$$ 3.2160.5

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)\times\mathrm{SU}(2)$$

## 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 14.a
Elliptic curve isogeny class 32.a

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

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

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

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

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

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an order of index $$8$$ in $$\Z \times \Z [\sqrt{-1}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-1})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$