# Properties

 Label 363.a.11979.1 Conductor $363$ Discriminant $-11979$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{10}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(33)$ by the Atkin-Lehner involution $w_3$, which has discriminant $3\cdot 11^9$.

## Simplified equation

 $y^2 + (x^2 + 1)y = x^5 + 2x^3 + 4x^2 + 2x$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = x^5z + 2x^3z^3 + 4x^2z^4 + 2xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + x^4 + 8x^3 + 18x^2 + 8x + 1$ (homogenize, minimize)

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

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

sage: X = HyperellipticCurve(R([1, 8, 18, 8, 1, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$363$$ $$=$$ $$3 \cdot 11^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-11979$$ $$=$$ $$- 3^{2} \cdot 11^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$344$$ $$=$$ $$2^{3} \cdot 43$$ $$I_4$$ $$=$$ $$-3068$$ $$=$$ $$- 2^{2} \cdot 13 \cdot 59$$ $$I_6$$ $$=$$ $$-526433$$ $$=$$ $$- 19 \cdot 103 \cdot 269$$ $$I_{10}$$ $$=$$ $$-47916$$ $$=$$ $$- 2^{2} \cdot 3^{2} \cdot 11^{3}$$ $$J_2$$ $$=$$ $$172$$ $$=$$ $$2^{2} \cdot 43$$ $$J_4$$ $$=$$ $$1744$$ $$=$$ $$2^{4} \cdot 109$$ $$J_6$$ $$=$$ $$45841$$ $$=$$ $$45841$$ $$J_8$$ $$=$$ $$1210779$$ $$=$$ $$3^{2} \cdot 29 \cdot 4639$$ $$J_{10}$$ $$=$$ $$-11979$$ $$=$$ $$- 3^{2} \cdot 11^{3}$$ $$g_1$$ $$=$$ $$-150536645632/11979$$ $$g_2$$ $$=$$ $$-8874253312/11979$$ $$g_3$$ $$=$$ $$-1356160144/11979$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (-1 : -34 : 4)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (-1 : -34 : 4)$$
All points: $$(1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 0 : 4)$$

magma: [C![-1,-34,4],C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model

magma: [C![-1,0,4],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,0]]; // simplified model

Number of rational Weierstrass points: $$3$$

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: $$\Z/{2}\Z \oplus \Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 5xz - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-15xz^2 - 6z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 5xz - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-15xz^2 - 6z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$x + z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 5xz - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - 30xz^2 - 11z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$18.97059$$ Tamagawa product: $$4$$ Torsion order: $$20$$ Leading coefficient: $$0.189705$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + T + 3 T^{2} )$$
$$11$$ $$2$$ $$3$$ $$2$$ $$( 1 - T )^{2}$$

## Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

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

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.120.3 yes
$$3$$ 3.80.4 no

## Sato-Tate group

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

## 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 11.a
Elliptic curve isogeny class 33.a

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$3$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);