↑

Properties

Label 147109.a.147109.1
Conductor $147109$
Discriminant $147109$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z\)
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = 4x^5 - 7x^4 + 4x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 4x^5z - 7x^4z^2 + 4x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 16x^5 - 28x^4 + 2x^3 + 16x^2 - 8x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 4, 0, -7, 4]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 4, 0, -7, 4], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -8, 16, 2, -28, 16, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(147109\) \(=\) \( 157 \cdot 937 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(147109\) \(=\) \( 157 \cdot 937 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(916\) \(=\)  \( 2^{2} \cdot 229 \)
\( I_4 \)  \(=\) \(23641\) \(=\)  \( 47 \cdot 503 \)
\( I_6 \)  \(=\) \(6548701\) \(=\)  \( 6548701 \)
\( I_{10} \)  \(=\) \(18829952\) \(=\)  \( 2^{7} \cdot 157 \cdot 937 \)
\( J_2 \)  \(=\) \(229\) \(=\)  \( 229 \)
\( J_4 \)  \(=\) \(1200\) \(=\)  \( 2^{4} \cdot 3 \cdot 5^{2} \)
\( J_6 \)  \(=\) \(-496\) \(=\)  \( - 2^{4} \cdot 31 \)
\( J_8 \)  \(=\) \(-388396\) \(=\)  \( - 2^{2} \cdot 89 \cdot 1091 \)
\( J_{10} \)  \(=\) \(147109\) \(=\)  \( 157 \cdot 937 \)
\( g_1 \)  \(=\) \(629763392149/147109\)
\( g_2 \)  \(=\) \(14410786800/147109\)
\( g_3 \)  \(=\) \(-26010736/147109\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((1 : -4 : 2)\)
\((1 : -5 : 2)\) \((-4 : -5 : 5)\) \((2 : -15 : 3)\) \((2 : -20 : 3)\) \((5 : -29 : 4)\) \((1 : -55 : 5)\)
\((-4 : -56 : 5)\) \((1 : -71 : 5)\) \((5 : -160 : 4)\) \((45 : -74240 : 44)\) \((45 : -102069 : 44)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((1 : -4 : 2)\)
\((1 : -5 : 2)\) \((-4 : -5 : 5)\) \((2 : -15 : 3)\) \((2 : -20 : 3)\) \((5 : -29 : 4)\) \((1 : -55 : 5)\)
\((-4 : -56 : 5)\) \((1 : -71 : 5)\) \((5 : -160 : 4)\) \((45 : -74240 : 44)\) \((45 : -102069 : 44)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 2)\)
\((1 : 1 : 2)\) \((2 : -5 : 3)\) \((2 : 5 : 3)\) \((1 : -16 : 5)\) \((1 : 16 : 5)\) \((-4 : -51 : 5)\)
\((-4 : 51 : 5)\) \((5 : -131 : 4)\) \((5 : 131 : 4)\) \((45 : -27829 : 44)\) \((45 : 27829 : 44)\)

magma: [C![-4,-56,5],C![-4,-5,5],C![0,-1,1],C![0,0,1],C![1,-71,5],C![1,-55,5],C![1,-5,2],C![1,-4,2],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-20,3],C![2,-15,3],C![5,-160,4],C![5,-29,4],C![45,-102069,44],C![45,-74240,44]]; // minimal model
 
magma: [C![-4,-51,5],C![-4,51,5],C![0,-1,1],C![0,1,1],C![1,-16,5],C![1,16,5],C![1,-1,2],C![1,1,2],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-5,3],C![2,5,3],C![5,-131,4],C![5,131,4],C![45,-27829,44],C![45,27829,44]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -5 : 2) - (1 : 0 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-2x^3 - z^3\) \(0.678271\) \(\infty\)
\((1 : -5 : 2) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(-3xz^2 - z^3\) \(0.578512\) \(\infty\)
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.193248\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -5 : 2) - (1 : 0 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-2x^3 - z^3\) \(0.678271\) \(\infty\)
\((1 : -5 : 2) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(-3xz^2 - z^3\) \(0.578512\) \(\infty\)
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.193248\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-3x^3 - z^3\) \(0.678271\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(x^3 - 6xz^2 - z^3\) \(0.578512\) \(\infty\)
\((0 : -1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.193248\) \(\infty\)

2-torsion field: 5.5.147109.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.062757 \)
Real period: \( 17.84814 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.120098 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(157\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 9 T + 157 T^{2} )\)
\(937\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 7 T + 937 T^{2} )\)

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);