Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -x^4 - x - 1$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -x^4z^2 - xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 2x^4 + x^2 - 4x - 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -1, 0, 0, -1]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -1, 0, 0, -1], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-4, -4, 1, 0, -2, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2722\) | \(=\) | \( 2 \cdot 1361 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(2722\) | \(=\) | \( 2 \cdot 1361 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(248\) | \(=\) | \( 2^{3} \cdot 31 \) |
\( I_4 \) | \(=\) | \(868\) | \(=\) | \( 2^{2} \cdot 7 \cdot 31 \) |
\( I_6 \) | \(=\) | \(71599\) | \(=\) | \( 11 \cdot 23 \cdot 283 \) |
\( I_{10} \) | \(=\) | \(10888\) | \(=\) | \( 2^{3} \cdot 1361 \) |
\( J_2 \) | \(=\) | \(124\) | \(=\) | \( 2^{2} \cdot 31 \) |
\( J_4 \) | \(=\) | \(496\) | \(=\) | \( 2^{4} \cdot 31 \) |
\( J_6 \) | \(=\) | \(1441\) | \(=\) | \( 11 \cdot 131 \) |
\( J_8 \) | \(=\) | \(-16833\) | \(=\) | \( - 3 \cdot 31 \cdot 181 \) |
\( J_{10} \) | \(=\) | \(2722\) | \(=\) | \( 2 \cdot 1361 \) |
\( g_1 \) | \(=\) | \(14658125312/1361\) | ||
\( g_2 \) | \(=\) | \(472842752/1361\) | ||
\( g_3 \) | \(=\) | \(11078408/1361\) |
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);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 1 : 1)\)
magma: [C![-1,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-1,0,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z/{5}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2\) | \(0\) | \(5\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 11.72001 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 5 \) |
Leading coefficient: | \( 0.468800 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 2 T^{2} )\) | |
\(1361\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 63 T + 1361 T^{2} )\) |
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 |
\(5\) | not computed | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
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);