Properties

Label 2848.a.45568.1
Conductor 2848
Discriminant 45568
Mordell-Weil group \(\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^6 - 2x^5 - 7x^4 - 5x^3 + x$ (homogenize, simplify)
$y^2 + z^3y = x^6 - 2x^5z - 7x^4z^2 - 5x^3z^3 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 8x^5 - 28x^4 - 20x^3 + 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -5, -7, -2, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -5, -7, -2, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, 4, 0, -20, -28, -8, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2848\) \(=\) \( 2^{5} \cdot 89 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(45568\) \(=\) \( 2^{9} \cdot 89 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(160\) \(=\)  \( 2^{5} \cdot 5 \)
\( I_4 \)  \(=\) \(38464\) \(=\)  \( 2^{6} \cdot 601 \)
\( I_6 \)  \(=\) \(702976\) \(=\)  \( 2^{9} \cdot 1373 \)
\( I_{10} \)  \(=\) \(186646528\) \(=\)  \( 2^{21} \cdot 89 \)
\( J_2 \)  \(=\) \(20\) \(=\)  \( 2^{2} \cdot 5 \)
\( J_4 \)  \(=\) \(-384\) \(=\)  \( - 2^{7} \cdot 3 \)
\( J_6 \)  \(=\) \(1024\) \(=\)  \( 2^{10} \)
\( J_8 \)  \(=\) \(-31744\) \(=\)  \( - 2^{10} \cdot 31 \)
\( J_{10} \)  \(=\) \(45568\) \(=\)  \( 2^{9} \cdot 89 \)
\( g_1 \)  \(=\) \(6250/89\)
\( g_2 \)  \(=\) \(-6000/89\)
\( g_3 \)  \(=\) \(800/89\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\)
\((-1 : -3 : 2)\) \((-1 : -5 : 2)\) \((-2 : -47 : 5)\) \((-2 : -78 : 5)\)

magma: [C![-2,-78,5],C![-2,-47,5],C![-1,-5,2],C![-1,-3,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0]];
 

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
\((-1 : -5 : 2) + (-1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x + z) (2x + z)\) \(=\) \(0,\) \(4y\) \(=\) \(3xz^2 - z^3\) \(0.002856\) \(\infty\)

2-torsion field: 6.2.182272.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.002856 \)
Real period: \( 19.02698 \)
Tamagawa product: \( 8 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.434815 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(9\) \(8\) \(1\)
\(89\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 7 T + 89 T^{2} )\)

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 \(\Q\).