Properties

Label 816.b.52224.1
Conductor 816
Discriminant -52224
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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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-17, 0, -27, 0, -12, 0, -1], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-17, 0, -27, 0, -12, 0, -1]), R([0, 1, 0, 1]))
 

$y^2 + (x^3 + x)y = -x^6 - 12x^4 - 27x^2 - 17$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 816 \)  =  \( 2^{4} \cdot 3 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-52224\)  =  \( -1 \cdot 2^{10} \cdot 3 \cdot 17 \)

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);
 

\( I_2 \)  =  \(-127712\)  =  \( -1 \cdot 2^{5} \cdot 13 \cdot 307 \)
\( I_4 \)  =  \(152372800\)  =  \( 2^{6} \cdot 5^{2} \cdot 95233 \)
\( I_6 \)  =  \(-5859681637888\)  =  \( -1 \cdot 2^{9} \cdot 13 \cdot 13523 \cdot 65101 \)
\( I_{10} \)  =  \(-213909504\)  =  \( -1 \cdot 2^{22} \cdot 3 \cdot 17 \)
\( J_2 \)  =  \(-15964\)  =  \( -1 \cdot 2^{2} \cdot 13 \cdot 307 \)
\( J_4 \)  =  \(9031504\)  =  \( 2^{4} \cdot 163 \cdot 3463 \)
\( J_6 \)  =  \(-6282991104\)  =  \( -1 \cdot 2^{9} \cdot 3 \cdot 13 \cdot 17 \cdot 83 \cdot 223 \)
\( J_8 \)  =  \(4683401370560\)  =  \( 2^{6} \cdot 5 \cdot 11 \cdot 139 \cdot 9572027 \)
\( J_{10} \)  =  \(-52224\)  =  \( -1 \cdot 2^{10} \cdot 3 \cdot 17 \)
\( g_1 \)  =  \(1012531723491160951/51\)
\( g_2 \)  =  \(35882713644370099/51\)
\( g_3 \)  =  \(30660536527816\)
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 $\R$.

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: 2.4237420772714490782030552961

Tamagawa numbers: 3 (p = 2), 1 (p = 3), 1 (p = 17)

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

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

2-torsion field: 8.0.95883264.1

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 34.a3
  Elliptic curve 24.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 \(\Q\).