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Properties

Label 62090.a.62090.1
Conductor $62090$
Discriminant $-62090$
Mordell-Weil group \(\Z \oplus \Z\)
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

$y^2 + (x + 1)y = x^5 - 4x^4 + x^3 + 6x^2 + 2x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z - 4x^4z^2 + x^3z^3 + 6x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 16x^4 + 4x^3 + 25x^2 + 10x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 6, 1, -4, 1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 6, 1, -4, 1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([1, 10, 25, 4, -16, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(62090\) \(=\) \( 2 \cdot 5 \cdot 7 \cdot 887 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-62090\) \(=\) \( - 2 \cdot 5 \cdot 7 \cdot 887 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2024\) \(=\)  \( 2^{3} \cdot 11 \cdot 23 \)
\( I_4 \)  \(=\) \(12352\) \(=\)  \( 2^{6} \cdot 193 \)
\( I_6 \)  \(=\) \(7932827\) \(=\)  \( 7 \cdot 1133261 \)
\( I_{10} \)  \(=\) \(-248360\) \(=\)  \( - 2^{3} \cdot 5 \cdot 7 \cdot 887 \)
\( J_2 \)  \(=\) \(1012\) \(=\)  \( 2^{2} \cdot 11 \cdot 23 \)
\( J_4 \)  \(=\) \(40614\) \(=\)  \( 2 \cdot 3 \cdot 7 \cdot 967 \)
\( J_6 \)  \(=\) \(2096441\) \(=\)  \( 19 \cdot 110339 \)
\( J_8 \)  \(=\) \(118025324\) \(=\)  \( 2^{2} \cdot 67 \cdot 440393 \)
\( J_{10} \)  \(=\) \(-62090\) \(=\)  \( - 2 \cdot 5 \cdot 7 \cdot 887 \)
\( g_1 \)  \(=\) \(-530728691964416/31045\)
\( g_2 \)  \(=\) \(-3006694244928/4435\)
\( g_3 \)  \(=\) \(-1073528735752/31045\)

  (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

Known points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (2 : 1 : 1),\, (2 : -4 : 1)\)
Known points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (2 : 1 : 1),\, (2 : -4 : 1)\)
Known points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (2 : -5 : 1),\, (2 : 5 : 1)\)

magma: [C![0,-1,1],C![0,0,1],C![1,0,0],C![2,-4,1],C![2,1,1]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,0,0],C![2,-5,1],C![2,5,1]]; // 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: \(\Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.337339\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.209747\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.337339\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.209747\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0.337339\) \(\infty\)
\((0 : 1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0.209747\) \(\infty\)

2-torsion field: 5.3.993440.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.059817 \)
Real period: \( 19.01639 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.137511 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 5 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 3 T + 7 T^{2} )\)
\(887\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 13 T + 887 T^{2} )\)

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

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);