Properties

Label 1632.a.52224.1
Conductor $1632$
Discriminant $52224$
Mordell-Weil group \(\Z/{12}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^6 + 11x^4 - 27x^2 + 17$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 + 11x^4z^2 - 27x^2z^4 + 17z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 46x^4 - 107x^2 + 68$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([17, 0, -27, 0, 11, 0, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![17, 0, -27, 0, 11, 0, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([68, 0, -107, 0, 46, 0, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1632\) \(=\) \( 2^{5} \cdot 3 \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(52224\) \(=\) \( 2^{10} \cdot 3 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(15964\) \(=\)  \( 2^{2} \cdot 13 \cdot 307 \)
\( I_4 \)  \(=\) \(2380825\) \(=\)  \( 5^{2} \cdot 95233 \)
\( I_6 \)  \(=\) \(11444690699\) \(=\)  \( 13 \cdot 13523 \cdot 65101 \)
\( I_{10} \)  \(=\) \(6528\) \(=\)  \( 2^{7} \cdot 3 \cdot 17 \)
\( J_2 \)  \(=\) \(15964\) \(=\)  \( 2^{2} \cdot 13 \cdot 307 \)
\( J_4 \)  \(=\) \(9031504\) \(=\)  \( 2^{4} \cdot 163 \cdot 3463 \)
\( J_6 \)  \(=\) \(6282991104\) \(=\)  \( 2^{9} \cdot 3 \cdot 13 \cdot 17 \cdot 83 \cdot 223 \)
\( J_8 \)  \(=\) \(4683401370560\) \(=\)  \( 2^{6} \cdot 5 \cdot 11 \cdot 139 \cdot 9572027 \)
\( J_{10} \)  \(=\) \(52224\) \(=\)  \( 2^{10} \cdot 3 \cdot 17 \)
\( g_1 \)  \(=\) \(1012531723491160951/51\)
\( g_2 \)  \(=\) \(35882713644370099/51\)
\( g_3 \)  \(=\) \(30660536527816\)

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(2 \cdot(1 : -2 : 1) - D_\infty\) \((x - z)^2\) \(=\) \(0,\) \(y\) \(=\) \(4xz^2 - 6z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\(2 \cdot(1 : -2 : 1) - D_\infty\) \((x - z)^2\) \(=\) \(0,\) \(y\) \(=\) \(4xz^2 - 6z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\(2 \cdot(1 : -2 : 1) - D_\infty\) \((x - z)^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 9xz^2 - 12z^3\) \(0\) \(12\)

2-torsion field: 8.8.1534132224.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 15.15713 \)
Tamagawa product: \( 6 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.631547 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(10\) \(6\) \(1 - T\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 17 T^{2} )\)

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?
\(2\) 2.45.1 yes
\(3\) 3.720.4 yes

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 48.a
  Elliptic curve isogeny class 34.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 \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);