Minimal equation
Minimal equation
Simplified equation
$y^2 = x^5 + 3x^4 - 5x^2 - x + 2$ | (homogenize, simplify) |
$y^2 = x^5z + 3x^4z^2 - 5x^2z^4 - xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^5 + 3x^4 - 5x^2 - x + 2$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -1, -5, 0, 3, 1]), R([]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -1, -5, 0, 3, 1], R![]);
sage: X = HyperellipticCurve(R([2, -1, -5, 0, 3, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(7680\) | \(=\) | \( 2^{9} \cdot 3 \cdot 5 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(46080\) | \(=\) | \( 2^{10} \cdot 3^{2} \cdot 5 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(200\) | \(=\) | \( 2^{3} \cdot 5^{2} \) |
\( I_4 \) | \(=\) | \(472\) | \(=\) | \( 2^{3} \cdot 59 \) |
\( I_6 \) | \(=\) | \(32580\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 181 \) |
\( I_{10} \) | \(=\) | \(180\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \) |
\( J_2 \) | \(=\) | \(400\) | \(=\) | \( 2^{4} \cdot 5^{2} \) |
\( J_4 \) | \(=\) | \(5408\) | \(=\) | \( 2^{5} \cdot 13^{2} \) |
\( J_6 \) | \(=\) | \(56320\) | \(=\) | \( 2^{10} \cdot 5 \cdot 11 \) |
\( J_8 \) | \(=\) | \(-1679616\) | \(=\) | \( - 2^{8} \cdot 3^{8} \) |
\( J_{10} \) | \(=\) | \(46080\) | \(=\) | \( 2^{10} \cdot 3^{2} \cdot 5 \) |
\( g_1 \) | \(=\) | \(2000000000/9\) | ||
\( g_2 \) | \(=\) | \(67600000/9\) | ||
\( g_3 \) | \(=\) | \(1760000/9\) |
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: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1),\, (-2 : 0 : 1)\)
magma: [C![-2,0,1],C![-1,0,1],C![1,0,0],C![1,0,1]]; // minimal model
magma: [C![-2,0,1],C![-1,0,1],C![1,0,0],C![1,0,1]]; // simplified model
Number of rational Weierstrass points: \(4\)
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: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 0 : 1) - (1 : 0 : 0)\) | \(x + 2z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-2 : 0 : 1) + (-1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 0 : 1) - (1 : 0 : 0)\) | \(x + 2z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-2 : 0 : 1) + (-1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 0 : 1) - (1 : 0 : 0)\) | \(x + 2z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-2 : 0 : 1) + (-1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{5}) \)
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 17.49721 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 1.093576 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(9\) | \(10\) | \(2\) | \(1\) | |
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 - 2 T + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.360.2 | 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);