# Properties

 Label 1272.a.122112.1 Conductor 1272 Discriminant -122112 Mordell-Weil group $$\Z/{2}\Z \times \Z/{6}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 2, 4, 3], R![1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 2, 4, 3]), R([1, 0, 1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 2, 4, 3], R![1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -4, -2, 8, 17, 12]))

 $y^2 + (x^2 + 1)y = 3x^5 + 4x^4 + 2x^3 - x^2 - x$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = 3x^5z + 4x^4z^2 + 2x^3z^3 - x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = 12x^5 + 17x^4 + 8x^3 - 2x^2 - 4x + 1$ (minimize, homogenize)

## Invariants

 $$N$$ = $$1272$$ = $$2^{3} \cdot 3 \cdot 53$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-122112$$ = $$- 2^{8} \cdot 3^{2} \cdot 53$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-992$$ = $$- 2^{5} \cdot 31$$ $$I_4$$ = $$-321728$$ = $$- 2^{6} \cdot 11 \cdot 457$$ $$I_6$$ = $$18153984$$ = $$2^{9} \cdot 3 \cdot 53 \cdot 223$$ $$I_{10}$$ = $$-500170752$$ = $$- 2^{20} \cdot 3^{2} \cdot 53$$ $$J_2$$ = $$-124$$ = $$- 2^{2} \cdot 31$$ $$J_4$$ = $$3992$$ = $$2^{3} \cdot 499$$ $$J_6$$ = $$79504$$ = $$2^{4} \cdot 4969$$ $$J_8$$ = $$-6448640$$ = $$- 2^{9} \cdot 5 \cdot 11 \cdot 229$$ $$J_{10}$$ = $$-122112$$ = $$- 2^{8} \cdot 3^{2} \cdot 53$$ $$g_1$$ = $$114516604/477$$ $$g_2$$ = $$29731418/477$$ $$g_3$$ = $$-4775209/477$$

## Automorphism group

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

## Rational points

magma: [C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,-15,3],C![1,0,0]];

Points: $$(0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (1 : -15 : 3)$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$3$$

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);

This curve is locally solvable everywhere.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{2}\Z \times \Z/{6}\Z$$

Generator Height Order
$$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$6$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$14.39791$$ Tamagawa product: $$4$$ Torsion order: $$12$$ Leading coefficient: $$0.399942$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$8$$ $$3$$ $$2$$ $$1 + T$$
$$3$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 - T + 3 T^{2} )$$
$$53$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 6 T + 53 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## 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 $$\Q$$.