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Properties

Label 100224.a.601344.1
Conductor $100224$
Discriminant $601344$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(100224\) \(=\) \( 2^{7} \cdot 3^{3} \cdot 29 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(601344\) \(=\) \( 2^{8} \cdot 3^{4} \cdot 29 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(72\) \(=\)  \( 2^{3} \cdot 3^{2} \)
\( I_4 \)  \(=\) \(648\) \(=\)  \( 2^{3} \cdot 3^{4} \)
\( I_6 \)  \(=\) \(17784\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 13 \cdot 19 \)
\( I_{10} \)  \(=\) \(2349\) \(=\)  \( 3^{4} \cdot 29 \)
\( J_2 \)  \(=\) \(144\) \(=\)  \( 2^{4} \cdot 3^{2} \)
\( J_4 \)  \(=\) \(-864\) \(=\)  \( - 2^{5} \cdot 3^{3} \)
\( J_6 \)  \(=\) \(-50432\) \(=\)  \( - 2^{8} \cdot 197 \)
\( J_8 \)  \(=\) \(-2002176\) \(=\)  \( - 2^{8} \cdot 3^{2} \cdot 11 \cdot 79 \)
\( J_{10} \)  \(=\) \(601344\) \(=\)  \( 2^{8} \cdot 3^{4} \cdot 29 \)
\( g_1 \)  \(=\) \(2985984/29\)
\( g_2 \)  \(=\) \(-124416/29\)
\( g_3 \)  \(=\) \(-50432/29\)

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

Copy content sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (1 : -3 : 1),\, (1 : 3 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -3 : 1),\, (1 : 3 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -3/2 : 1),\, (1 : 3/2 : 1)\)

Copy content magma:[C![1,-3,1],C![1,0,0],C![1,3,1]]; // minimal model
 
Copy content magma:[C![1,-3/2,1],C![1,0,0],C![1,3/2,1]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

Copy content magma:#Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

Copy content 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: \(\Z \oplus \Z/{2}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-3z^3\) \(0.501426\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-3z^3\) \(0.501426\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -3/2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-3/2z^3\) \(0.501426\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 6.0.22707.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.501426 \)
Real period: \( 5.342534 \)
Tamagawa product: \( 4 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.678888 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(7\) \(8\) \(2\) \(1^*\) \(1\) no
\(3\) \(3\) \(4\) \(2\) \(-1\) \(1 - T\) yes
\(29\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 29 T^{2} )\) yes

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 \(\overline{\Q}\)

Copy content 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 \(\Q\).

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

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

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