Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = -x^6 + x^5 - 5x^4 + 14x^3 - 12x^2 + 27x - 38$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = -x^6 + x^5z - 5x^4z^2 + 14x^3z^3 - 12x^2z^4 + 27xz^5 - 38z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 + 4x^5 - 19x^4 + 58x^3 - 45x^2 + 110x - 151$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-38, 27, -12, 14, -5, 1, -1]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-38, 27, -12, 14, -5, 1, -1], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([-151, 110, -45, 58, -19, 4, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(449012\) | \(=\) | \( 2^{2} \cdot 112253 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-898024\) | \(=\) | \( - 2^{3} \cdot 112253 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(60428\) | \(=\) | \( 2^{2} \cdot 15107 \) |
\( I_4 \) | \(=\) | \(225918169\) | \(=\) | \( 225918169 \) |
\( I_6 \) | \(=\) | \(3418701707075\) | \(=\) | \( 5^{2} \cdot 136748068283 \) |
\( I_{10} \) | \(=\) | \(114947072\) | \(=\) | \( 2^{10} \cdot 112253 \) |
\( J_2 \) | \(=\) | \(15107\) | \(=\) | \( 15107 \) |
\( J_4 \) | \(=\) | \(95970\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 457 \) |
\( J_6 \) | \(=\) | \(602044\) | \(=\) | \( 2^{2} \cdot 41 \cdot 3671 \) |
\( J_8 \) | \(=\) | \(-28790548\) | \(=\) | \( - 2^{2} \cdot 19 \cdot 378823 \) |
\( J_{10} \) | \(=\) | \(898024\) | \(=\) | \( 2^{3} \cdot 112253 \) |
\( g_1 \) | \(=\) | \(786848544941745592307/898024\) | ||
\( g_2 \) | \(=\) | \(165439872520613355/449012\) | ||
\( g_3 \) | \(=\) | \(34349838510439/224506\) |
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
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\R$.
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: \(\Z/{3}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2 + 5z^3\) | \(0\) | \(3\) |
2-torsion field: 6.0.57473536.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 1.363736 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 3.636630 \) |
Analytic order of Ш: | \( 8 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T )\) | |
\(112253\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 126 T + 112253 T^{2} )\) |
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? |
---|---|---|
\(3\) | 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 \(\overline{\Q}\)
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);