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Properties

Label 1280.a.12800.1
Conductor $1280$
Discriminant $-12800$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{6}\Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(1280\) \(=\) \( 2^{8} \cdot 5 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-12800\) \(=\) \( - 2^{9} \cdot 5^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(22\) \(=\)  \( 2 \cdot 11 \)
\( I_4 \)  \(=\) \(-170\) \(=\)  \( - 2 \cdot 5 \cdot 17 \)
\( I_6 \)  \(=\) \(-1832\) \(=\)  \( - 2^{3} \cdot 229 \)
\( I_{10} \)  \(=\) \(-50\) \(=\)  \( - 2 \cdot 5^{2} \)
\( J_2 \)  \(=\) \(44\) \(=\)  \( 2^{2} \cdot 11 \)
\( J_4 \)  \(=\) \(534\) \(=\)  \( 2 \cdot 3 \cdot 89 \)
\( J_6 \)  \(=\) \(7684\) \(=\)  \( 2^{2} \cdot 17 \cdot 113 \)
\( J_8 \)  \(=\) \(13235\) \(=\)  \( 5 \cdot 2647 \)
\( J_{10} \)  \(=\) \(-12800\) \(=\)  \( - 2^{9} \cdot 5^{2} \)
\( g_1 \)  \(=\) \(-322102/25\)
\( g_2 \)  \(=\) \(-355377/100\)
\( g_3 \)  \(=\) \(-232441/200\)

  (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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -4 : 2)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -4 : 2)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 2)\)

Copy content magma:[C![0,-1,1],C![0,0,1],C![1,-4,2],C![1,0,0]]; // minimal model
 
Copy content magma:[C![0,-1,1],C![0,1,1],C![1,0,2],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

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

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

2-torsion field: \(\Q(\zeta_{8})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 17.07078 \)
Tamagawa product: \( 4 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.474188 \)
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?
\(2\) \(8\) \(9\) \(2\) \(-1^*\) \(1 + 2 T^{2}\) no
\(5\) \(1\) \(2\) \(2\) \(-1\) \(( 1 - T )( 1 + 5 T^{2} )\) 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?
\(2\) 2.180.3 yes
\(3\) 3.80.1 yes

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);