# Properties

 Label 277.a.277.2 Conductor $277$ Discriminant $277$ Mordell-Weil group $$\Z/{5}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + y = x^5 - 9x^4 + 14x^3 - 19x^2 + 11x - 6$ (homogenize, simplify) $y^2 + z^3y = x^5z - 9x^4z^2 + 14x^3z^3 - 19x^2z^4 + 11xz^5 - 6z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 36x^4 + 56x^3 - 76x^2 + 44x - 23$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, 11, -19, 14, -9, 1]), R([1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, 11, -19, 14, -9, 1], R![1]);

sage: X = HyperellipticCurve(R([-23, 44, -76, 56, -36, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$277$$ $$=$$ $$277$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$277$$ $$=$$ $$277$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$4480$$ $$=$$ $$2^{7} \cdot 5 \cdot 7$$ $$I_4$$ $$=$$ $$1370512$$ $$=$$ $$2^{4} \cdot 11 \cdot 13 \cdot 599$$ $$I_6$$ $$=$$ $$1511819744$$ $$=$$ $$2^{5} \cdot 337 \cdot 140191$$ $$I_{10}$$ $$=$$ $$-1108$$ $$=$$ $$- 2^{2} \cdot 277$$ $$J_2$$ $$=$$ $$2240$$ $$=$$ $$2^{6} \cdot 5 \cdot 7$$ $$J_4$$ $$=$$ $$-19352$$ $$=$$ $$- 2^{3} \cdot 41 \cdot 59$$ $$J_6$$ $$=$$ $$164384$$ $$=$$ $$2^{5} \cdot 11 \cdot 467$$ $$J_8$$ $$=$$ $$-1569936$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 32707$$ $$J_{10}$$ $$=$$ $$-277$$ $$=$$ $$-277$$ $$g_1$$ $$=$$ $$-56394933862400000/277$$ $$g_2$$ $$=$$ $$217505333248000/277$$ $$g_3$$ $$=$$ $$-824813158400/277$$

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

## Rational points

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

magma: [C![1,0,0]]; // minimal model

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

Number of rational Weierstrass points: $$1$$

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: $$\Z/{5}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$3.578416$$ Tamagawa product: $$1$$ Torsion order: $$5$$ Leading coefficient: $$0.143136$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$277$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 8 T + 277 T^{2} )$$

## 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
$$2$$ 2.6.1
$$3$$ 3.80.2

## 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 $$\Q$$.