Properties

Label 909.a.8181.1
Conductor $909$
Discriminant $8181$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{8}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = 3x^5 - 7x^4 + x^3 + 6x^2 - 3x$ (homogenize, simplify)
$y^2 + xz^2y = 3x^5z - 7x^4z^2 + x^3z^3 + 6x^2z^4 - 3xz^5$ (dehomogenize, simplify)
$y^2 = 12x^5 - 28x^4 + 4x^3 + 25x^2 - 12x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 6, 1, -7, 3]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 6, 1, -7, 3], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, -12, 25, 4, -28, 12]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(909\) \(=\) \( 3^{2} \cdot 101 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(8181\) \(=\) \( 3^{4} \cdot 101 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1384\) \(=\)  \( 2^{3} \cdot 173 \)
\( I_4 \)  \(=\) \(44560\) \(=\)  \( 2^{4} \cdot 5 \cdot 557 \)
\( I_6 \)  \(=\) \(19431635\) \(=\)  \( 5 \cdot 71 \cdot 127 \cdot 431 \)
\( I_{10} \)  \(=\) \(32724\) \(=\)  \( 2^{2} \cdot 3^{4} \cdot 101 \)
\( J_2 \)  \(=\) \(692\) \(=\)  \( 2^{2} \cdot 173 \)
\( J_4 \)  \(=\) \(12526\) \(=\)  \( 2 \cdot 6263 \)
\( J_6 \)  \(=\) \(35569\) \(=\)  \( 35569 \)
\( J_8 \)  \(=\) \(-33071732\) \(=\)  \( - 2^{2} \cdot 17 \cdot 486349 \)
\( J_{10} \)  \(=\) \(8181\) \(=\)  \( 3^{4} \cdot 101 \)
\( g_1 \)  \(=\) \(158683025503232/8181\)
\( g_2 \)  \(=\) \(4150789321088/8181\)
\( g_3 \)  \(=\) \(17032713616/8181\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (4 : -18 : 3)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (4 : -18 : 3)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1),\, (4 : 0 : 3)\)

magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![4,-18,3]]; // minimal model
 
magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,1,1],C![4,0,3]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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

2-torsion field: 3.3.404.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 21.80554 \)
Tamagawa product: \( 4 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.340711 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(2\) \(4\) \(4\) \(( 1 - T )( 1 + T )\)
\(101\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 6 T + 101 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).