Minimal equation
Minimal equation
Simplified equation
$y^2 + y = -x^6 + 3x^5 - 50x^4 + 95x^3 - 14x^2 - 33x - 6$ | (homogenize, simplify) |
$y^2 + z^3y = -x^6 + 3x^5z - 50x^4z^2 + 95x^3z^3 - 14x^2z^4 - 33xz^5 - 6z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 + 12x^5 - 200x^4 + 380x^3 - 56x^2 - 132x - 23$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, -33, -14, 95, -50, 3, -1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, -33, -14, 95, -50, 3, -1], R![1]);
sage: X = HyperellipticCurve(R([-23, -132, -56, 380, -200, 12, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1083\) | \(=\) | \( 3 \cdot 19^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-390963\) | \(=\) | \( - 3 \cdot 19^{4} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(150440\) | \(=\) | \( 2^{3} \cdot 5 \cdot 3761 \) |
\( I_4 \) | \(=\) | \(1945515892\) | \(=\) | \( 2^{2} \cdot 486378973 \) |
\( I_6 \) | \(=\) | \(68956865081488\) | \(=\) | \( 2^{4} \cdot 17 \cdot 97 \cdot 2857 \cdot 914801 \) |
\( I_{10} \) | \(=\) | \(-1563852\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 19^{4} \) |
\( J_2 \) | \(=\) | \(75220\) | \(=\) | \( 2^{2} \cdot 5 \cdot 3761 \) |
\( J_4 \) | \(=\) | \(-88500632\) | \(=\) | \( - 2^{3} \cdot 11 \cdot 19 \cdot 41 \cdot 1291 \) |
\( J_6 \) | \(=\) | \(98386538568\) | \(=\) | \( 2^{3} \cdot 3 \cdot 19^{3} \cdot 597673 \) |
\( J_8 \) | \(=\) | \(-107931608328616\) | \(=\) | \( - 2^{3} \cdot 19^{2} \cdot 37 \cdot 1010065961 \) |
\( J_{10} \) | \(=\) | \(-390963\) | \(=\) | \( - 3 \cdot 19^{4} \) |
\( g_1 \) | \(=\) | \(-2408056349828975363200000/390963\) | ||
\( g_2 \) | \(=\) | \(1982406707133537344000/20577\) | ||
\( g_3 \) | \(=\) | \(-27053302090985600/19\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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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: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 0.132918 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.265837 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) | |
\(19\) | \(2\) | \(4\) | \(1\) | \(( 1 - T )( 1 + T )\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.15.2 | no |
\(3\) | 3.720.5 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 19.a
Elliptic curve isogeny class 57.b
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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);