Properties

Label 1008.a.27216.1
Conductor 1008
Discriminant 27216
Mordell-Weil group \(\Z/{2}\Z \times \Z/{8}\Z\)
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

$y^2 + (x^3 + x)y = -4x^4 + 15x^2 - 21$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -4x^4z^2 + 15x^2z^4 - 21z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 14x^4 + 61x^2 - 84$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-21, 0, 15, 0, -4], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-21, 0, 15, 0, -4]), R([0, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-84, 0, 61, 0, -14, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(1008\) = \( 2^{4} \cdot 3^{2} \cdot 7 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(27216\) = \( 2^{4} \cdot 3^{5} \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(33824\) =  \( 2^{5} \cdot 7 \cdot 151 \)
\( I_4 \)  = \(151936\) =  \( 2^{7} \cdot 1187 \)
\( I_6 \)  = \(1707222272\) =  \( 2^{8} \cdot 7 \cdot 952691 \)
\( I_{10} \)  = \(111476736\) =  \( 2^{16} \cdot 3^{5} \cdot 7 \)
\( J_2 \)  = \(4228\) =  \( 2^{2} \cdot 7 \cdot 151 \)
\( J_4 \)  = \(743250\) =  \( 2 \cdot 3 \cdot 5^{3} \cdot 991 \)
\( J_6 \)  = \(173847744\) =  \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 43117 \)
\( J_8 \)  = \(45651924783\) =  \( 3^{3} \cdot 59 \cdot 191 \cdot 150041 \)
\( J_{10} \)  = \(27216\) =  \( 2^{4} \cdot 3^{5} \cdot 7 \)
\( g_1 \)  = \(12063042849801664/243\)
\( g_2 \)  = \(167186257609000/81\)
\( g_3 \)  = \(3083035208512/27\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1)\)

magma: [C![-2,5,1],C![1,-1,0],C![1,0,0],C![2,-5,1]];
 

Number of rational Weierstrass points: \(2\)

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

This curve is locally solvable everywhere.

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);
 

Mordell-Weil group of the Jacobian:

Group structure: \(\Z/{2}\Z \times \Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 5 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - 2z) (x + 2z)\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 2z^3\) \(0\) \(8\)

2-torsion field: \(\Q(\sqrt{3}, \sqrt{7})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 12.16748 \)
Tamagawa product: \( 8 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.380233 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(4\) \(1\) \(1 + T + 2 T^{2}\)
\(3\) \(5\) \(2\) \(8\) \(( 1 - T )^{2}\)
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 7 T^{2} )\)

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 21.a5
  Elliptic curve 48.a4

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 \(\Q\).