Properties

Label 672.a.172032.1
Conductor 672
Discriminant 172032
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

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![56, 0, -75, 0, -16, 0, -1], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([56, 0, -75, 0, -16, 0, -1]), R([0, 1, 0, 1]))
 

$y^2 + (x^3 + x)y = -x^6 - 16x^4 - 75x^2 + 56$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(672,2),R![1, 1]>*])); Factorization($1);
 
\( N \)  =  \( 672 \)  =  \( 2^{5} \cdot 3 \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(172032\)  =  \( 2^{13} \cdot 3 \cdot 7 \)

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);
 

\( I_2 \)  =  \(-135328\)  =  \( -1 \cdot 2^{5} \cdot 4229 \)
\( I_4 \)  =  \(9671540800\)  =  \( 2^{6} \cdot 5^{2} \cdot 6044713 \)
\( I_6 \)  =  \(-119230681068032\)  =  \( -1 \cdot 2^{9} \cdot 397 \cdot 586580413 \)
\( I_{10} \)  =  \(704643072\)  =  \( 2^{25} \cdot 3 \cdot 7 \)
\( J_2 \)  =  \(-16916\)  =  \( -1 \cdot 2^{2} \cdot 4229 \)
\( J_4 \)  =  \(-88822256\)  =  \( -1 \cdot 2^{4} \cdot 103 \cdot 53897 \)
\( J_6 \)  =  \(-277597802496\)  =  \( -1 \cdot 2^{12} \cdot 3 \cdot 7 \cdot 3227281 \)
\( J_8 \)  =  \(-798387183476800\)  =  \( -1 \cdot 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829 \)
\( J_{10} \)  =  \(172032\)  =  \( 2^{13} \cdot 3 \cdot 7 \)
\( g_1 \)  =  \(-1352659309173012149/168\)
\( g_2 \)  =  \(419870026410625699/168\)
\( g_3 \)  =  \(-461744933079368\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

Rational points

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 $\Q_{2}$.

magma: [];
 

There are no rational points.

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
 

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: twice a square

Regulator: 1.0

Real period: 1.1133493018149438309175533043

Tamagawa numbers: 2 (p = 2), 1 (p = 3), 1 (p = 7)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

Torsion: \(\Z/{4}\Z\)

2-torsion field: 8.0.199148544.2

Sato-Tate group

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

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 48.a1
  Elliptic curve 14.a1

Endomorphisms

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 \(\Q\).