Properties

Label 336.a.172032.1
Conductor 336
Discriminant -172032
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-56, 0, -75, 0, 15, 0, -1], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-56, 0, -75, 0, 15, 0, -1]), R([0, 1, 0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-56, 0, -75, 0, 15, 0, -1], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-224, 0, -299, 0, 62, 0, -3]))
 

$y^2 + (x^3 + x)y = -x^6 + 15x^4 - 75x^2 - 56$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 + 15x^4z^2 - 75x^2z^4 - 56z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 62x^4 - 299x^2 - 224$ (minimize, homogenize)

Invariants

\( N \)  =  \(336\) = \( 2^{4} \cdot 3 \cdot 7 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(336,2),R![1, 1]>*])); Factorization($1);
 
\( \Delta \)  =  \(-172032\) = \( - 2^{13} \cdot 3 \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(135328\) =  \( 2^{5} \cdot 4229 \)
\( I_4 \)  = \(9671540800\) =  \( 2^{6} \cdot 5^{2} \cdot 6044713 \)
\( I_6 \)  = \(119230681068032\) =  \( 2^{9} \cdot 397 \cdot 586580413 \)
\( I_{10} \)  = \(-704643072\) =  \( - 2^{25} \cdot 3 \cdot 7 \)
\( J_2 \)  = \(16916\) =  \( 2^{2} \cdot 4229 \)
\( J_4 \)  = \(-88822256\) =  \( - 2^{4} \cdot 103 \cdot 53897 \)
\( J_6 \)  = \(277597802496\) =  \( 2^{12} \cdot 3 \cdot 7 \cdot 3227281 \)
\( J_8 \)  = \(-798387183476800\) =  \( - 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829 \)
\( J_{10} \)  = \(-172032\) =  \( - 2^{13} \cdot 3 \cdot 7 \)
\( g_1 \)  = \(-1352659309173012149/168\)
\( g_2 \)  = \(419870026410625699/168\)
\( g_3 \)  = \(-461744933079368\)

Automorphism group

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

Rational points

magma: [];
 

This curve has no rational points.

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

Number of rational Weierstrass points: \(0\)

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);
 

This curve is locally solvable except over $\Q_{2}$.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{2}\Z\)

Generator Height Order
\(3x^2 - 32z^2\) \(=\) \(0,\) \(6y\) \(=\) \(-35xz^2\) \(0\) \(2\)

2-torsion field: 8.4.260112384.2

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 0.356066 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.178033 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(13\) \(4\) \(1\) \(1 + T\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 7 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 14.a1
  Elliptic curve 24.a1

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 \(\Q\).