Properties

Label 10048.a.10048.1
Conductor $10048$
Discriminant $-10048$
Mordell-Weil group \(\Z\)
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

$y^2 + (x^3 + x^2)y = 2x^4 - x^3 + x^2 - 4x + 2$ (homogenize, simplify)
$y^2 + (x^3 + x^2z)y = 2x^4z^2 - x^3z^3 + x^2z^4 - 4xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 9x^4 - 4x^3 + 4x^2 - 16x + 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -4, 1, -1, 2]), R([0, 0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -4, 1, -1, 2], R![0, 0, 1, 1]);
 
sage: X = HyperellipticCurve(R([8, -16, 4, -4, 9, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10048\) \(=\) \( 2^{6} \cdot 157 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-10048\) \(=\) \( - 2^{6} \cdot 157 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(460\) \(=\)  \( 2^{2} \cdot 5 \cdot 23 \)
\( I_4 \)  \(=\) \(16375\) \(=\)  \( 5^{3} \cdot 131 \)
\( I_6 \)  \(=\) \(1900672\) \(=\)  \( 2^{7} \cdot 31 \cdot 479 \)
\( I_{10} \)  \(=\) \(1256\) \(=\)  \( 2^{3} \cdot 157 \)
\( J_2 \)  \(=\) \(460\) \(=\)  \( 2^{2} \cdot 5 \cdot 23 \)
\( J_4 \)  \(=\) \(-2100\) \(=\)  \( - 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
\( J_6 \)  \(=\) \(-69264\) \(=\)  \( - 2^{4} \cdot 3^{2} \cdot 13 \cdot 37 \)
\( J_8 \)  \(=\) \(-9067860\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 5 \cdot 50377 \)
\( J_{10} \)  \(=\) \(10048\) \(=\)  \( 2^{6} \cdot 157 \)
\( g_1 \)  \(=\) \(321817150000/157\)
\( g_2 \)  \(=\) \(-3193837500/157\)
\( g_3 \)  \(=\) \(-229004100/157\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -2 : 1),\, (7 : -95 : 8),\, (7 : -640 : 8)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -2 : 1),\, (7 : -95 : 8),\, (7 : -640 : 8)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (1 : -2 : 1),\, (1 : 2 : 1),\, (7 : -545 : 8),\, (7 : 545 : 8)\)

magma: [C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![7,-640,8],C![7,-95,8]]; // minimal model
 
magma: [C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![7,-545,8],C![7,545,8]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.041489\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.041489\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + x^2z - 2z^3\) \(0.041489\) \(\infty\)

2-torsion field: 6.0.160768.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.041489 \)
Real period: \( 17.15923 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.711923 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(6\) \(1\) \(1 + T\)
\(157\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 7 T + 157 T^{2} )\)

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 \(\Q\).