Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = 15x^5 + 50x^4 + 55x^3 + 22x^2 + 3x$ | (homogenize, simplify) |
$y^2 + xz^2y = 15x^5z + 50x^4z^2 + 55x^3z^3 + 22x^2z^4 + 3xz^5$ | (dehomogenize, simplify) |
$y^2 = 60x^5 + 200x^4 + 220x^3 + 89x^2 + 12x$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 22, 55, 50, 15]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, 22, 55, 50, 15], R![0, 1]);
sage: X = HyperellipticCurve(R([0, 12, 89, 220, 200, 60]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1125\) | \(=\) | \( 3^{2} \cdot 5^{3} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-151875\) | \(=\) | \( - 3^{5} \cdot 5^{4} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(8600\) | \(=\) | \( 2^{3} \cdot 5^{2} \cdot 43 \) |
\( I_4 \) | \(=\) | \(612100\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 6121 \) |
\( I_6 \) | \(=\) | \(1556297975\) | \(=\) | \( 5^{2} \cdot 62251919 \) |
\( I_{10} \) | \(=\) | \(-607500\) | \(=\) | \( - 2^{2} \cdot 3^{5} \cdot 5^{4} \) |
\( J_2 \) | \(=\) | \(4300\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 43 \) |
\( J_4 \) | \(=\) | \(668400\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 557 \) |
\( J_6 \) | \(=\) | \(132975225\) | \(=\) | \( 3^{2} \cdot 5^{2} \cdot 37 \cdot 15973 \) |
\( J_8 \) | \(=\) | \(31258726875\) | \(=\) | \( 3^{3} \cdot 5^{4} \cdot 211 \cdot 8779 \) |
\( J_{10} \) | \(=\) | \(-151875\) | \(=\) | \( - 3^{5} \cdot 5^{4} \) |
\( g_1 \) | \(=\) | \(-2352135088000000/243\) | ||
\( g_2 \) | \(=\) | \(-28342655360000/81\) | ||
\( g_3 \) | \(=\) | \(-437104339600/27\) |
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
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 18 : 3)\)
magma: [C![-4,18,3],C![0,0,1],C![1,0,0]]; // minimal model
magma: [C![-4,0,3],C![0,0,1],C![1,0,0]]; // simplified model
Number of rational Weierstrass points: \(3\)
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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-4 : 18 : 3) - (1 : 0 : 0)\) | \(3x + 4z\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(2z^3\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-4 : 18 : 3) - (1 : 0 : 0)\) | \(3x + 4z\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(2z^3\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x + 4z\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(xz^2 + 4z^3\) | \(0\) | \(2\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 1.964401 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.491100 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(2\) | \(5\) | \(2\) | \(( 1 + T )^{2}\) | |
\(5\) | \(3\) | \(4\) | \(2\) | \(1 - T\) |
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? |
---|---|---|
\(2\) | 2.120.3 | yes |
\(3\) | 3.80.4 | 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 15.a
Elliptic curve isogeny class 75.c
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(3\) 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);