Properties

Label 3456.e.442368.1
Conductor $3456$
Discriminant $-442368$
Mordell-Weil group \(\Z/{12}\Z\)
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

$y^2 + x^3y = -x^4 - 3x^2 + 12$ (homogenize, simplify)
$y^2 + x^3y = -x^4z^2 - 3x^2z^4 + 12z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^4 - 12x^2 + 48$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([12, 0, -3, 0, -1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![12, 0, -3, 0, -1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([48, 0, -12, 0, -4, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3456\) \(=\) \( 2^{7} \cdot 3^{3} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(3456,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-442368\) \(=\) \( - 2^{14} \cdot 3^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(384\) \(=\)  \( 2^{7} \cdot 3 \)
\( I_4 \)  \(=\) \(2295\) \(=\)  \( 3^{3} \cdot 5 \cdot 17 \)
\( I_6 \)  \(=\) \(331704\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 17 \cdot 271 \)
\( I_{10} \)  \(=\) \(54\) \(=\)  \( 2 \cdot 3^{3} \)
\( J_2 \)  \(=\) \(1536\) \(=\)  \( 2^{9} \cdot 3 \)
\( J_4 \)  \(=\) \(73824\) \(=\)  \( 2^{5} \cdot 3 \cdot 769 \)
\( J_6 \)  \(=\) \(-36864\) \(=\)  \( - 2^{12} \cdot 3^{2} \)
\( J_8 \)  \(=\) \(-1376651520\) \(=\)  \( - 2^{8} \cdot 3^{2} \cdot 5 \cdot 73 \cdot 1637 \)
\( J_{10} \)  \(=\) \(442368\) \(=\)  \( 2^{14} \cdot 3^{3} \)
\( g_1 \)  \(=\) \(19327352832\)
\( g_2 \)  \(=\) \(604766208\)
\( g_3 \)  \(=\) \(-196608\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 4 : 1),\, (2 : -4 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 4 : 1),\, (2 : -4 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : 0 : 1),\, (2 : 0 : 1)\)

magma: [C![-2,4,1],C![1,-1,0],C![1,0,0],C![2,-4,1]]; // minimal model
 
magma: [C![-2,0,1],C![1,-1,0],C![1,1,0],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: \(\Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((2 : -4 : 1) - (1 : 0 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + 4z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\((2 : -4 : 1) - (1 : 0 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + 4z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\((2 : 0 : 1) - (1 : 1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + 8z^3\) \(0\) \(12\)

2-torsion field: 4.0.432.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 16.84431 \)
Tamagawa product: \( 6 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.701846 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(7\) \(14\) \(6\) \(1\)
\(3\) \(3\) \(3\) \(1\) \(1 + T\)

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 96.a
  Elliptic curve isogeny class 36.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);