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Properties

Label 14899.b.14899.1
Conductor $14899$
Discriminant $-14899$
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

$y^2 + y = x^5 - x^4 - 3x^3 + 4x^2 - x$ (homogenize, simplify)
$y^2 + z^3y = x^5z - x^4z^2 - 3x^3z^3 + 4x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 4x^4 - 12x^3 + 16x^2 - 4x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 4, -3, -1, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 4, -3, -1, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, -4, 16, -12, -4, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(14899\) \(=\) \( 47 \cdot 317 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-14899\) \(=\) \( - 47 \cdot 317 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(312\) \(=\)  \( 2^{3} \cdot 3 \cdot 13 \)
\( I_4 \)  \(=\) \(5904\) \(=\)  \( 2^{4} \cdot 3^{2} \cdot 41 \)
\( I_6 \)  \(=\) \(428760\) \(=\)  \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 397 \)
\( I_{10} \)  \(=\) \(-59596\) \(=\)  \( - 2^{2} \cdot 47 \cdot 317 \)
\( J_2 \)  \(=\) \(156\) \(=\)  \( 2^{2} \cdot 3 \cdot 13 \)
\( J_4 \)  \(=\) \(30\) \(=\)  \( 2 \cdot 3 \cdot 5 \)
\( J_6 \)  \(=\) \(3788\) \(=\)  \( 2^{2} \cdot 947 \)
\( J_8 \)  \(=\) \(147507\) \(=\)  \( 3 \cdot 49169 \)
\( J_{10} \)  \(=\) \(-14899\) \(=\)  \( - 47 \cdot 317 \)
\( g_1 \)  \(=\) \(-92389579776/14899\)
\( g_2 \)  \(=\) \(-113892480/14899\)
\( g_3 \)  \(=\) \(-92184768/14899\)

  (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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((-1 : 2 : 1)\)
\((2 : 2 : 1)\) \((-1 : -3 : 1)\) \((2 : -3 : 1)\) \((5 : -30 : 4)\) \((5 : -34 : 4)\)
Known points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((-1 : 2 : 1)\)
\((2 : 2 : 1)\) \((-1 : -3 : 1)\) \((2 : -3 : 1)\) \((5 : -30 : 4)\) \((5 : -34 : 4)\)
Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\) \((-1 : -5 : 1)\)
\((-1 : 5 : 1)\) \((2 : -5 : 1)\) \((2 : 5 : 1)\) \((5 : -4 : 4)\) \((5 : 4 : 4)\)

magma: [C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-3,1],C![2,2,1],C![5,-34,4],C![5,-30,4]]; // minimal model
 
magma: [C![-1,-5,1],C![-1,5,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,0,0],C![1,1,1],C![2,-5,1],C![2,5,1],C![5,-4,4],C![5,4,4]]; // 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
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.285368\) \(\infty\)
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.075182\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.285368\) \(\infty\)
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.075182\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.285368\) \(\infty\)
\((1 : 1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.075182\) \(\infty\)

2-torsion field: 5.3.238384.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.020821 \)
Real period: \( 25.08777 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.522377 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(47\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 47 T^{2} )\)
\(317\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 317 T^{2} )\)

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);