# Properties

 Label 1312.c.671744.1 Conductor 1312 Discriminant -671744 Mordell-Weil group $$\Z/{22}\Z$$ 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

 $y^2 + (x + 1)y = x^6 + 4x^5 + 7x^4 + 5x^3 + 2x^2$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^6 + 4x^5z + 7x^4z^2 + 5x^3z^3 + 2x^2z^4$ (dehomogenize, simplify) $y^2 = 4x^6 + 16x^5 + 28x^4 + 20x^3 + 9x^2 + 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 2, 5, 7, 4, 1], R![1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 2, 5, 7, 4, 1]), R([1, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 2, 9, 20, 28, 16, 4]))

## Invariants

 Conductor: $$N$$ = $$1312$$ = $$2^{5} \cdot 41$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1312,2),R![1, -1]>*])); Factorization($1); Discriminant: $$\Delta$$ = $$-671744$$ = $$- 2^{14} \cdot 41$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-1312$$ = $$- 2^{5} \cdot 41$$ $$I_4$$ = $$92224$$ = $$2^{6} \cdot 11 \cdot 131$$ $$I_6$$ = $$-29946368$$ = $$- 2^{9} \cdot 23 \cdot 2543$$ $$I_{10}$$ = $$-2751463424$$ = $$- 2^{26} \cdot 41$$ $$J_2$$ = $$-164$$ = $$- 2^{2} \cdot 41$$ $$J_4$$ = $$160$$ = $$2^{5} \cdot 5$$ $$J_6$$ = $$-1984$$ = $$- 2^{6} \cdot 31$$ $$J_8$$ = $$74944$$ = $$2^{6} \cdot 1171$$ $$J_{10}$$ = $$-671744$$ = $$- 2^{14} \cdot 41$$ $$g_1$$ = $$2825761/16$$ $$g_2$$ = $$8405/8$$ $$g_3$$ = $$1271/16$$

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

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

## Automorphism group

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

## Rational points

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

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

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: $$\Z/{22}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$22$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$11.85781$$ Tamagawa product: $$22$$ Torsion order: $$22$$ Leading coefficient: $$0.538991$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$14$$ $$5$$ $$22$$ $$1 - T$$
$$41$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 41 T^{2} )$$

## 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 $$\Q$$.