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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

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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 $2$.

Minimal equation

Minimal equation

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)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 2, 2, 1]), R([1, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 2, 2, 1], R![1, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, 2, 5, 10, 10, 4, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

Copy content sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_6$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
Copy content 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: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : -1 : 1),\, (-1 : 1 : 1)\)

Copy content 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
 
Copy content 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: \(0\)

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

This curve is locally solvable everywhere.

Copy content 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: \(\Z/{21}\Z\)

Copy content 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
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 - z^3\) \(0\) \(21\)

2-torsion field: 6.0.41472.1

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 Root number* L-factor Cluster picture Tame reduction?
\(2\) \(2\) \(3\) \(3\) \(1^*\) \(1 + T + T^{2}\) yes
\(3\) \(4\) \(4\) \(1\) \(1^*\) \(1 + 3 T + 3 T^{2}\) no

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

Copy content 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 \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

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

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

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