Properties

 Label 324.a.648.1 Conductor $324$ Discriminant $-648$ Mordell-Weil group $$\Z/{21}\Z$$ Sato-Tate group $E_3$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

Related objects

Show commands: Magma / SageMath

This is a model for the modular curve $X_1(18)$. The integer $18$ is the largest $N \in \mathbb{N}$ such that $X_1(N)$ has genus $2$.

Simplified equation

 $y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + 2x^3 + x^2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = x^5z + 2x^4z^2 + 2x^3z^3 + x^2z^4$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1$ (homogenize, minimize)

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

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

sage: X = HyperellipticCurve(R([1, 2, 5, 10, 10, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$324$$ $$=$$ $$2^{2} \cdot 3^{4}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-648$$ $$=$$ $$- 2^{3} \cdot 3^{4}$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$60$$ $$=$$ $$2^{2} \cdot 3 \cdot 5$$ $$I_4$$ $$=$$ $$945$$ $$=$$ $$3^{3} \cdot 5 \cdot 7$$ $$I_6$$ $$=$$ $$2295$$ $$=$$ $$3^{3} \cdot 5 \cdot 17$$ $$I_{10}$$ $$=$$ $$82944$$ $$=$$ $$2^{10} \cdot 3^{4}$$ $$J_2$$ $$=$$ $$15$$ $$=$$ $$3 \cdot 5$$ $$J_4$$ $$=$$ $$-30$$ $$=$$ $$- 2 \cdot 3 \cdot 5$$ $$J_6$$ $$=$$ $$140$$ $$=$$ $$2^{2} \cdot 5 \cdot 7$$ $$J_8$$ $$=$$ $$300$$ $$=$$ $$2^{2} \cdot 3 \cdot 5^{2}$$ $$J_{10}$$ $$=$$ $$648$$ $$=$$ $$2^{3} \cdot 3^{4}$$ $$g_1$$ $$=$$ $$9375/8$$ $$g_2$$ $$=$$ $$-625/4$$ $$g_3$$ $$=$$ $$875/18$$

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

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

Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_6$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_6$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Rational points

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

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

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

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: $$\Z/{21}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 - z^3$$ $$0$$ $$21$$
Generator $D_0$ Height Order
$$(-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 3xz^2 - z^3$$ $$0$$ $$21$$

BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$25.52176$$ Tamagawa product: $$3$$ Torsion order: $$21$$ Leading coefficient: $$0.173617$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$3$$ $$4$$ $$4$$ $$1$$ $$1 + 3 T + 3 T^{2}$$

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?
$$2$$ 2.40.3 no
$$3$$ 3.1920.3 yes

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_3$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial:
$$x^{3} - 3 x - 1$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 3.3.81.1-8.1-a

magma: HeuristicDecompositionFactors(C);

Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial $$x^{3} - 3 x - 1$$

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

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

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

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);