Properties

Label 3564.a.128304.1
Conductor $3564$
Discriminant $-128304$
Mordell-Weil group \(\Z \oplus \Z/{6}\Z\)
Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{CM} \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -2x^4 - x^2 + 11$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -2x^4z^2 - x^2z^4 + 11z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 6x^4 - 3x^2 + 44$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([11, 0, -1, 0, -2]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![11, 0, -1, 0, -2], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([44, 0, -3, 0, -6, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3564\) \(=\) \( 2^{2} \cdot 3^{4} \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-128304\) \(=\) \( - 2^{4} \cdot 3^{6} \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(904\) \(=\)  \( 2^{3} \cdot 113 \)
\( I_4 \)  \(=\) \(15840\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \)
\( I_6 \)  \(=\) \(5450316\) \(=\)  \( 2^{2} \cdot 3 \cdot 67 \cdot 6779 \)
\( I_{10} \)  \(=\) \(2112\) \(=\)  \( 2^{6} \cdot 3 \cdot 11 \)
\( J_2 \)  \(=\) \(1356\) \(=\)  \( 2^{2} \cdot 3 \cdot 113 \)
\( J_4 \)  \(=\) \(52854\) \(=\)  \( 2 \cdot 3 \cdot 23 \cdot 383 \)
\( J_6 \)  \(=\) \(-1629760\) \(=\)  \( - 2^{6} \cdot 5 \cdot 11 \cdot 463 \)
\( J_8 \)  \(=\) \(-1250874969\) \(=\)  \( - 3 \cdot 347 \cdot 911 \cdot 1319 \)
\( J_{10} \)  \(=\) \(128304\) \(=\)  \( 2^{4} \cdot 3^{6} \cdot 11 \)
\( g_1 \)  \(=\) \(1179158514752/33\)
\( g_2 \)  \(=\) \(101683837384/99\)
\( g_3 \)  \(=\) \(-1891855040/81\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : -2 : 1)\) \((1 : 2 : 1)\) \((-1 : 4 : 1)\) \((1 : -4 : 1)\)
\((-2 : 5 : 1)\) \((2 : -5 : 1)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : -2 : 1)\) \((1 : 2 : 1)\) \((-1 : 4 : 1)\) \((1 : -4 : 1)\)
\((-2 : 5 : 1)\) \((2 : -5 : 1)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-2 : 0 : 1)\) \((2 : 0 : 1)\) \((-1 : -6 : 1)\) \((-1 : 6 : 1)\)
\((1 : -6 : 1)\) \((1 : 6 : 1)\)

magma: [C![-2,5,1],C![-1,-2,1],C![-1,4,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![2,-5,1]]; // minimal model
 
magma: [C![-2,0,1],C![-1,-6,1],C![-1,6,1],C![1,-6,1],C![1,-1,0],C![1,1,0],C![1,6,1],C![2,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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: \(\Z \oplus \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + xz - 7z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + 2z^3\) \(0.075642\) \(\infty\)
\((-1 : 4 : 1) + (1 : 2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + 3z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + xz - 7z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 + 2z^3\) \(0.075642\) \(\infty\)
\((-1 : 4 : 1) + (1 : 2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + 3z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + xz - 7z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 + 4z^3\) \(0.075642\) \(\infty\)
\((-1 : 6 : 1) + (1 : 6 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - xz^2 + 6z^3\) \(0\) \(6\)

2-torsion field: 4.0.5808.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.075642 \)
Real period: \( 18.92295 \)
Tamagawa product: \( 12 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.477128 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(3\) \(1 + T + 2 T^{2}\)
\(3\) \(4\) \(6\) \(4\) \(1\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 11 T^{2} )\)

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?
\(2\) 2.90.3 yes
\(3\) 3.720.4 yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

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 36.a
  Elliptic curve isogeny class 99.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 \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(8\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)

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);