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Properties

Label 2528.a.161792.1
Conductor $2528$
Discriminant $161792$
Mordell-Weil group \(\Z \oplus \Z/{4}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = 8x^5 + 27x^4 + 25x^3 - x^2 - 6x + 1$ (homogenize, simplify)
$y^2 + xz^2y = 8x^5z + 27x^4z^2 + 25x^3z^3 - x^2z^4 - 6xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 32x^5 + 108x^4 + 100x^3 - 3x^2 - 24x + 4$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -6, -1, 25, 27, 8]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -6, -1, 25, 27, 8], R![0, 1]);
 
sage: X = HyperellipticCurve(R([4, -24, -3, 100, 108, 32]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2528\) \(=\) \( 2^{5} \cdot 79 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(161792\) \(=\) \( 2^{11} \cdot 79 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(4308\) \(=\)  \( 2^{2} \cdot 3 \cdot 359 \)
\( I_4 \)  \(=\) \(41769\) \(=\)  \( 3^{3} \cdot 7 \cdot 13 \cdot 17 \)
\( I_6 \)  \(=\) \(57202227\) \(=\)  \( 3^{3} \cdot 2118601 \)
\( I_{10} \)  \(=\) \(20224\) \(=\)  \( 2^{8} \cdot 79 \)
\( J_2 \)  \(=\) \(4308\) \(=\)  \( 2^{2} \cdot 3 \cdot 359 \)
\( J_4 \)  \(=\) \(745440\) \(=\)  \( 2^{5} \cdot 3 \cdot 5 \cdot 1553 \)
\( J_6 \)  \(=\) \(167549072\) \(=\)  \( 2^{4} \cdot 10471817 \)
\( J_8 \)  \(=\) \(41530152144\) \(=\)  \( 2^{4} \cdot 3 \cdot 13 \cdot 127 \cdot 524053 \)
\( J_{10} \)  \(=\) \(161792\) \(=\)  \( 2^{11} \cdot 79 \)
\( g_1 \)  \(=\) \(1449033801989157/158\)
\( g_2 \)  \(=\) \(29101128101235/79\)
\( g_3 \)  \(=\) \(12146564220993/632\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 1 : 1),\, (-5 : 36 : 4),\, (-5 : 44 : 4)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 1 : 1),\, (-5 : 36 : 4),\, (-5 : 44 : 4)\)
All points: \((1 : 0 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (0 : -2 : 1),\, (0 : 2 : 1),\, (-5 : -8 : 4),\, (-5 : 8 : 4)\)

magma: [C![-5,36,4],C![-5,44,4],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,0,0]]; // minimal model
 
magma: [C![-5,-8,4],C![-5,8,4],C![-1,-1,1],C![-1,1,1],C![0,-2,1],C![0,2,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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: \(\Z \oplus \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.081850\) \(\infty\)
\((-5 : 44 : 4) + (0 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (4x + 5z)\) \(=\) \(0,\) \(4y\) \(=\) \(xz^2 + 4z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.081850\) \(\infty\)
\((-5 : 44 : 4) + (0 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (4x + 5z)\) \(=\) \(0,\) \(4y\) \(=\) \(xz^2 + 4z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.081850\) \(\infty\)
\((0 : 2 : 1) - (1 : 0 : 0)\) \(x (4x + 5z)\) \(=\) \(0,\) \(4y\) \(=\) \(3xz^2 + 8z^3\) \(0\) \(4\)

2-torsion field: 6.6.12781568.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.081850 \)
Real period: \( 17.46310 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.357341 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(11\) \(4\) \(1 + T\)
\(79\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 79 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.60.1 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);