Properties

Label 20172.b.968256.1
Conductor 20172
Discriminant -968256
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![1, -3, 0, 4, -2, -1, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -3, 0, 4, -2, -1, 1]), R([0, 1, 1]))
 

$y^2 + (x^2 + x)y = x^6 - x^5 - 2x^4 + 4x^3 - 3x + 1$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 20172 \)  =  \( 2^{2} \cdot 3 \cdot 41^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-968256\)  =  \( -1 \cdot 2^{6} \cdot 3^{2} \cdot 41^{2} \)

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);
 

\( I_2 \)  =  \(136\)  =  \( 2^{3} \cdot 17 \)
\( I_4 \)  =  \(774916\)  =  \( 2^{2} \cdot 23 \cdot 8423 \)
\( I_6 \)  =  \(-219321592\)  =  \( -1 \cdot 2^{3} \cdot 7 \cdot 613 \cdot 6389 \)
\( I_{10} \)  =  \(-3965976576\)  =  \( -1 \cdot 2^{18} \cdot 3^{2} \cdot 41^{2} \)
\( J_2 \)  =  \(17\)  =  \( 17 \)
\( J_4 \)  =  \(-8060\)  =  \( -1 \cdot 2^{2} \cdot 5 \cdot 13 \cdot 31 \)
\( J_6 \)  =  \(418896\)  =  \( 2^{4} \cdot 3^{2} \cdot 2909 \)
\( J_8 \)  =  \(-14460592\)  =  \( -1 \cdot 2^{4} \cdot 83 \cdot 10889 \)
\( J_{10} \)  =  \(-968256\)  =  \( -1 \cdot 2^{6} \cdot 3^{2} \cdot 41^{2} \)
\( g_1 \)  =  \(-1419857/968256\)
\( g_2 \)  =  \(9899695/242064\)
\( g_3 \)  =  \(-840701/6724\)
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 everywhere.

magma: [C![-3,-11,2],C![-3,5,2],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-9,1],C![2,3,1],C![3,-68,5],C![3,-52,5]];
 

Known rational points: (-3 : -11 : 2), (-3 : 5 : 2), (-1 : 0 : 1), (0 : -1 : 1), (0 : 1 : 1), (1 : -3 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 1), (1 : 1 : 0), (2 : -9 : 1), (2 : 3 : 1), (3 : -68 : 5), (3 : -52 : 5)

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

Number of rational Weierstrass points: \(2\)

Invariants of the Jacobian:

Analytic rank*: \(2\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.0178319893453

Real period: 16.482903876584195628030077693

Tamagawa numbers: 4 (p = 2), 2 (p = 3), 1 (p = 41)

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

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

2-torsion field: 4.0.656.1

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 246.a1
  Elliptic curve 82.a2

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 \(\Q\).