Properties

Label 1176.b.16464.1
Conductor $1176$
Discriminant $16464$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{6}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = -2x^5 + x^2$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = -2x^5z + x^2z^4$ (dehomogenize, simplify)
$y^2 = -8x^5 + 5x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 0, 0, -2]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 0, 0, -2], R![1, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, 5, 0, 0, -8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1176\) \(=\) \( 2^{3} \cdot 3 \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(16464\) \(=\) \( 2^{4} \cdot 3 \cdot 7^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(160\) \(=\)  \( 2^{5} \cdot 5 \)
\( I_4 \)  \(=\) \(4720\) \(=\)  \( 2^{4} \cdot 5 \cdot 59 \)
\( I_6 \)  \(=\) \(130020\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 197 \)
\( I_{10} \)  \(=\) \(-65856\) \(=\)  \( - 2^{6} \cdot 3 \cdot 7^{3} \)
\( J_2 \)  \(=\) \(80\) \(=\)  \( 2^{4} \cdot 5 \)
\( J_4 \)  \(=\) \(-520\) \(=\)  \( - 2^{3} \cdot 5 \cdot 13 \)
\( J_6 \)  \(=\) \(4220\) \(=\)  \( 2^{2} \cdot 5 \cdot 211 \)
\( J_8 \)  \(=\) \(16800\) \(=\)  \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \)
\( J_{10} \)  \(=\) \(-16464\) \(=\)  \( - 2^{4} \cdot 3 \cdot 7^{3} \)
\( g_1 \)  \(=\) \(-204800000/1029\)
\( g_2 \)  \(=\) \(16640000/1029\)
\( g_3 \)  \(=\) \(-1688000/1029\)

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

Automorphism group

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

Rational points

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

magma: [C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0]]; // simplified model
 

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: \(\Z/{2}\Z \times \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0\) \(2\)
\((0 : 1 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0\) \(6\)

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: \( 13.01930 \)
Tamagawa product: \( 4 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.361647 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 14.a
  Elliptic curve isogeny class 84.b

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(3\) 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 \(\Q\).