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Properties

Label 10681.a.117491.1
Conductor $10681$
Discriminant $-117491$
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

$y^2 + y = x^5 - 23x^3 + 69x^2 - 77x + 30$ (homogenize, simplify)
$y^2 + z^3y = x^5z - 23x^3z^3 + 69x^2z^4 - 77xz^5 + 30z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 92x^3 + 276x^2 - 308x + 121$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([30, -77, 69, -23, 0, 1]), R([1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![30, -77, 69, -23, 0, 1], R![1]);
 
Copy content sage:X = HyperellipticCurve(R([121, -308, 276, -92, 0, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10681\) \(=\) \( 11 \cdot 971 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-117491\) \(=\) \( - 11^{2} \cdot 971 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(376\) \(=\)  \( 2^{3} \cdot 47 \)
\( I_4 \)  \(=\) \(-7952\) \(=\)  \( - 2^{4} \cdot 7 \cdot 71 \)
\( I_6 \)  \(=\) \(-514680\) \(=\)  \( - 2^{3} \cdot 3 \cdot 5 \cdot 4289 \)
\( I_{10} \)  \(=\) \(-469964\) \(=\)  \( - 2^{2} \cdot 11^{2} \cdot 971 \)
\( J_2 \)  \(=\) \(188\) \(=\)  \( 2^{2} \cdot 47 \)
\( J_4 \)  \(=\) \(2798\) \(=\)  \( 2 \cdot 1399 \)
\( J_6 \)  \(=\) \(3356\) \(=\)  \( 2^{2} \cdot 839 \)
\( J_8 \)  \(=\) \(-1799469\) \(=\)  \( - 3^{3} \cdot 7 \cdot 9521 \)
\( J_{10} \)  \(=\) \(-117491\) \(=\)  \( - 11^{2} \cdot 971 \)
\( g_1 \)  \(=\) \(-234849287168/117491\)
\( g_2 \)  \(=\) \(-18591792256/117491\)
\( g_3 \)  \(=\) \(-118614464/117491\)

  (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

Known points
\((1 : 0 : 0)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((2 : 0 : 1)\) \((2 : -1 : 1)\) \((0 : 5 : 1)\)
\((0 : -6 : 1)\) \((3 : 6 : 1)\) \((3 : -7 : 1)\) \((-3 : 35 : 1)\) \((-3 : -36 : 1)\)
Known points
\((1 : 0 : 0)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((2 : 0 : 1)\) \((2 : -1 : 1)\) \((0 : 5 : 1)\)
\((0 : -6 : 1)\) \((3 : 6 : 1)\) \((3 : -7 : 1)\) \((-3 : 35 : 1)\) \((-3 : -36 : 1)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((0 : -11 : 1)\)
\((0 : 11 : 1)\) \((3 : -13 : 1)\) \((3 : 13 : 1)\) \((-3 : -71 : 1)\) \((-3 : 71 : 1)\)

Copy content magma:[C![-3,-36,1],C![-3,35,1],C![0,-6,1],C![0,5,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-1,1],C![2,0,1],C![3,-7,1],C![3,6,1]]; // minimal model
 
Copy content magma:[C![-3,-71,1],C![-3,71,1],C![0,-11,1],C![0,11,1],C![1,-1,1],C![1,0,0],C![1,1,1],C![2,-1,1],C![2,1,1],C![3,-13,1],C![3,13,1]]; // 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: \(\Z \oplus \Z\)

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

Generator $D_0$ Height Order
\((0 : -6 : 1) + (2 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(3xz^2 - 6z^3\) \(0.309259\) \(\infty\)
\((0 : -6 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(5xz^2 - 6z^3\) \(0.068272\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -6 : 1) + (2 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(3xz^2 - 6z^3\) \(0.309259\) \(\infty\)
\((0 : -6 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(5xz^2 - 6z^3\) \(0.068272\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -11 : 1) + (2 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(6xz^2 - 11z^3\) \(0.309259\) \(\infty\)
\((0 : -11 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(10xz^2 - 11z^3\) \(0.068272\) \(\infty\)

2-torsion field: 5.3.15536.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.020227 \)
Real period: \( 10.93482 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.442376 \)
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?
\(11\) \(1\) \(2\) \(2\) \(-1\) \(( 1 - T )( 1 + 5 T + 11 T^{2} )\) yes
\(971\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 51 T + 971 T^{2} )\) 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);