Properties

Label 456960.c.913920.1
Conductor 456960
Discriminant -913920
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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-714, 0, -413, 0, -79, 0, -5], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-714, 0, -413, 0, -79, 0, -5]), R([0, 1]))
 

$y^2 + xy = -5x^6 - 79x^4 - 413x^2 - 714$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 456960 \)  =  \( 2^{8} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-913920\)  =  \( -1 \cdot 2^{9} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)

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);
 

\( I_2 \)  =  \(-22056256\)  =  \( -1 \cdot 2^{6} \cdot 344629 \)
\( I_4 \)  =  \(932935552\)  =  \( 2^{7} \cdot 7288559 \)
\( I_6 \)  =  \(-6849525368597504\)  =  \( -1 \cdot 2^{10} \cdot 37 \cdot 180783503183 \)
\( I_{10} \)  =  \(-3743416320\)  =  \( -1 \cdot 2^{21} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
\( J_2 \)  =  \(-2757032\)  =  \( -1 \cdot 2^{3} \cdot 344629 \)
\( J_4 \)  =  \(316708008964\)  =  \( 2^{2} \cdot 563 \cdot 140634107 \)
\( J_6 \)  =  \(-48506712556968960\)  =  \( -1 \cdot 2^{10} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 859 \cdot 3253 \cdot 9497 \)
\( J_8 \)  =  \(8357648948106035343356\)  =  \( 2^{2} \cdot 19 \cdot 41 \cdot 2682172319674594141 \)
\( J_{10} \)  =  \(-913920\)  =  \( -1 \cdot 2^{9} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
\( g_1 \)  =  \(311127982838559560387716745536/1785\)
\( g_2 \)  =  \(12963268178085404526178374196/1785\)
\( g_3 \)  =  \(403438438743571080790912\)
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 $\R$ and $\Q_{2}$.

magma: [];
 

There are no rational points.

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(1\)

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

2-Selmer rank: \(6\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.519810465616

Real period: 1.5874538464207562876384391100

Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 7), 1 (p = 17)

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

Torsion: \(\Z/{2}\Z\)

2-torsion field: splitting field of \(x^{8} - 4 x^{7} + 66 x^{6} - 184 x^{5} + 1197 x^{4} - 2092 x^{3} + 4616 x^{2} - 3600 x + 3394\) with Galois group $D_4\times C_2$

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 714.a1
  Elliptic curve 640.h1

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 \(\Q\).