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Properties

Label 279259.a.279259.1
Conductor $279259$
Discriminant $-279259$
Mordell-Weil group \(\Z/{5}\Z\)
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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Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^5 + 9x^4 + 30x^3 + 28x^2 + 5x$ (homogenize, simplify)
$y^2 + z^3y = x^5z + 9x^4z^2 + 30x^3z^3 + 28x^2z^4 + 5xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 36x^4 + 120x^3 + 112x^2 + 20x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 5, 28, 30, 9, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 5, 28, 30, 9, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, 20, 112, 120, 36, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(279259\) \(=\) \( 17 \cdot 16427 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-279259\) \(=\) \( - 17 \cdot 16427 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(6272\) \(=\)  \( 2^{7} \cdot 7^{2} \)
\( I_4 \)  \(=\) \(-164576\) \(=\)  \( - 2^{5} \cdot 37 \cdot 139 \)
\( I_6 \)  \(=\) \(-183316832\) \(=\)  \( - 2^{5} \cdot 5728651 \)
\( I_{10} \)  \(=\) \(-1117036\) \(=\)  \( - 2^{2} \cdot 17 \cdot 16427 \)
\( J_2 \)  \(=\) \(3136\) \(=\)  \( 2^{6} \cdot 7^{2} \)
\( J_4 \)  \(=\) \(437200\) \(=\)  \( 2^{4} \cdot 5^{2} \cdot 1093 \)
\( J_6 \)  \(=\) \(67865696\) \(=\)  \( 2^{5} \cdot 31 \cdot 37 \cdot 43^{2} \)
\( J_8 \)  \(=\) \(5420745664\) \(=\)  \( 2^{6} \cdot 17 \cdot 307 \cdot 16229 \)
\( J_{10} \)  \(=\) \(-279259\) \(=\)  \( - 17 \cdot 16427 \)
\( g_1 \)  \(=\) \(-303305489096114176/279259\)
\( g_2 \)  \(=\) \(-13483676218163200/279259\)
\( g_3 \)  \(=\) \(-667424915849216/279259\)

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

magma: [C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],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: \(\Z/{5}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(5xz^2\) \(0\) \(5\)
Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(5xz^2\) \(0\) \(5\)
Generator $D_0$ Height Order
\((0 : 1 : 1) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(10xz^2 + z^3\) \(0\) \(5\)

2-torsion field: 5.3.4468144.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 7.247794 \)
Tamagawa product: \( 1 \)
Torsion order:\( 5 \)
Leading coefficient: \( 2.609206 \)
Analytic order of Ш: \( 9 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 3 T + 17 T^{2} )\)
\(16427\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 132 T + 16427 T^{2} )\)

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?
\(2\) 2.6.1 no
\(5\) not computed 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);