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Properties

Label 100227.a.902043.1
Conductor $100227$
Discriminant $-902043$
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^5 + x^4 + 23x^2 + 57x + 36$ (homogenize, simplify)
$y^2 + z^3y = x^5z + x^4z^2 + 23x^2z^4 + 57xz^5 + 36z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 4x^4 + 92x^2 + 228x + 145$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([36, 57, 23, 0, 1, 1]), R([1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![36, 57, 23, 0, 1, 1], R![1]);
 
Copy content sage:X = HyperellipticCurve(R([145, 228, 92, 0, 4, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(100227\) \(=\) \( 3 \cdot 33409 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-902043\) \(=\) \( - 3^{3} \cdot 33409 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(7648\) \(=\)  \( 2^{5} \cdot 239 \)
\( I_4 \)  \(=\) \(-11120\) \(=\)  \( - 2^{4} \cdot 5 \cdot 139 \)
\( I_6 \)  \(=\) \(-25359504\) \(=\)  \( - 2^{4} \cdot 3 \cdot 37 \cdot 109 \cdot 131 \)
\( I_{10} \)  \(=\) \(-3608172\) \(=\)  \( - 2^{2} \cdot 3^{3} \cdot 33409 \)
\( J_2 \)  \(=\) \(3824\) \(=\)  \( 2^{4} \cdot 239 \)
\( J_4 \)  \(=\) \(611144\) \(=\)  \( 2^{3} \cdot 79 \cdot 967 \)
\( J_6 \)  \(=\) \(130289488\) \(=\)  \( 2^{4} \cdot 7 \cdot 751 \cdot 1549 \)
\( J_8 \)  \(=\) \(31182503344\) \(=\)  \( 2^{4} \cdot 29 \cdot 181 \cdot 371291 \)
\( J_{10} \)  \(=\) \(-902043\) \(=\)  \( - 3^{3} \cdot 33409 \)
\( g_1 \)  \(=\) \(-817691377217306624/902043\)
\( g_2 \)  \(=\) \(-34174109226336256/902043\)
\( g_3 \)  \(=\) \(-1905220056076288/902043\)

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

Copy content magma:[C![-1,-2,1],C![-1,1,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,-3,1],C![-1,3,1],C![1,0,0]]; // 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: \(\Z\)

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

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - 6z^2\) \(=\) \(0,\) \(y\) \(=\) \(7xz^2 + 11z^3\) \(0.051324\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - 6z^2\) \(=\) \(0,\) \(y\) \(=\) \(7xz^2 + 11z^3\) \(0.051324\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - 6z^2\) \(=\) \(0,\) \(y\) \(=\) \(14xz^2 + 23z^3\) \(0.051324\) \(\infty\)

2-torsion field: 5.3.1603632.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.051324 \)
Real period: \( 7.555720 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.163379 \)
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?
\(3\) \(1\) \(3\) \(3\) \(-1\) \(( 1 - T )( 1 + T + 3 T^{2} )\) yes
\(33409\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 126 T + 33409 T^{2} )\) 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?
\(2\) 2.6.1 no

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);