Properties

Label 960.a.983040.1
Conductor 960
Discriminant -983040
Mordell-Weil group \(\Z/{2}\Z \times \Z/{4}\Z\)
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

$y^2 = x^5 - 2x^4 - x^3 - 2x^2 + x$ (homogenize, simplify)
$y^2 = x^5z - 2x^4z^2 - x^3z^3 - 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = x^5 - 2x^4 - x^3 - 2x^2 + x$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -2, -1, -2, 1], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -2, -1, -2, 1]), R([]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([0, 1, -2, -1, -2, 1]))
 

Invariants

Conductor: \( N \)  =  \(960\) = \( 2^{6} \cdot 3 \cdot 5 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(960,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-983040\) = \( - 2^{16} \cdot 3 \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-288\) =  \( - 2^{5} \cdot 3^{2} \)
\( I_4 \)  = \(33792\) =  \( 2^{10} \cdot 3 \cdot 11 \)
\( I_6 \)  = \(-21823488\) =  \( - 2^{16} \cdot 3^{2} \cdot 37 \)
\( I_{10} \)  = \(-4026531840\) =  \( - 2^{28} \cdot 3 \cdot 5 \)
\( J_2 \)  = \(-36\) =  \( - 2^{2} \cdot 3^{2} \)
\( J_4 \)  = \(-298\) =  \( - 2 \cdot 149 \)
\( J_6 \)  = \(34260\) =  \( 2^{2} \cdot 3 \cdot 5 \cdot 571 \)
\( J_8 \)  = \(-330541\) =  \( - 43 \cdot 7687 \)
\( J_{10} \)  = \(-983040\) =  \( - 2^{16} \cdot 3 \cdot 5 \)
\( g_1 \)  = \(19683/320\)
\( g_2 \)  = \(-36207/2560\)
\( g_3 \)  = \(-46251/1024\)

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: \((1 : 0 : 0),\, (0 : 0 : 1)\)

magma: [C![0,0,1],C![1,0,0]];
 

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: \(\Z/{2}\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 5xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(8xz^2 - 2z^3\) \(0\) \(4\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 6.402317 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.400144 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(16\) \(6\) \(4\) \(1\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)

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 40.a2
  Elliptic curve 24.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 \(\Q\).