Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^5 + 2x^4 - 2x^3 - x^2$ | (homogenize, simplify) |
$y^2 + z^3y = x^5z + 2x^4z^2 - 2x^3z^3 - x^2z^4$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 8x^4 - 8x^3 - 4x^2 + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, -2, 2, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, -2, 2, 1], R![1]);
sage: X = HyperellipticCurve(R([1, 0, -4, -8, 8, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(160507\) | \(=\) | \( 160507 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-160507\) | \(=\) | \( -160507 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(224\) | \(=\) | \( 2^{5} \cdot 7 \) |
\( I_4 \) | \(=\) | \(3856\) | \(=\) | \( 2^{4} \cdot 241 \) |
\( I_6 \) | \(=\) | \(161264\) | \(=\) | \( 2^{4} \cdot 10079 \) |
\( I_{10} \) | \(=\) | \(-642028\) | \(=\) | \( - 2^{2} \cdot 160507 \) |
\( J_2 \) | \(=\) | \(112\) | \(=\) | \( 2^{4} \cdot 7 \) |
\( J_4 \) | \(=\) | \(-120\) | \(=\) | \( - 2^{3} \cdot 3 \cdot 5 \) |
\( J_6 \) | \(=\) | \(5328\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 37 \) |
\( J_8 \) | \(=\) | \(145584\) | \(=\) | \( 2^{4} \cdot 3^{3} \cdot 337 \) |
\( J_{10} \) | \(=\) | \(-160507\) | \(=\) | \( -160507 \) |
\( g_1 \) | \(=\) | \(-17623416832/160507\) | ||
\( g_2 \) | \(=\) | \(168591360/160507\) | ||
\( g_3 \) | \(=\) | \(-66834432/160507\) |
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
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -2 : 1)\) | \((-2 : 3 : 1)\) | \((-2 : -4 : 1)\) | \((1 : -6 : 4)\) | \((3 : 18 : 1)\) | \((3 : -19 : 1)\) |
\((1 : -58 : 4)\) | \((6 : 99 : 1)\) | \((6 : -100 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -2 : 1)\) | \((-2 : 3 : 1)\) | \((-2 : -4 : 1)\) | \((1 : -6 : 4)\) | \((3 : 18 : 1)\) | \((3 : -19 : 1)\) |
\((1 : -58 : 4)\) | \((6 : 99 : 1)\) | \((6 : -100 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((-1 : -3 : 1)\) |
\((-1 : 3 : 1)\) | \((-2 : -7 : 1)\) | \((-2 : 7 : 1)\) | \((3 : -37 : 1)\) | \((3 : 37 : 1)\) | \((1 : -52 : 4)\) |
\((1 : 52 : 4)\) | \((6 : -199 : 1)\) | \((6 : 199 : 1)\) |
magma: [C![-2,-4,1],C![-2,3,1],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-58,4],C![1,-6,4],C![1,-1,1],C![1,0,0],C![1,0,1],C![3,-19,1],C![3,18,1],C![6,-100,1],C![6,99,1]]; // minimal model
magma: [C![-2,-7,1],C![-2,7,1],C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-52,4],C![1,52,4],C![1,-1,1],C![1,0,0],C![1,1,1],C![3,-37,1],C![3,37,1],C![6,-199,1],C![6,199,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.709131\) | \(\infty\) |
\((-1 : -2 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.865919\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.135793\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.709131\) | \(\infty\) |
\((-1 : -2 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.865919\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.135793\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.709131\) | \(\infty\) |
\((-1 : -3 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3z^3\) | \(0.865919\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.135793\) | \(\infty\) |
2-torsion field: 5.3.2568112.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.067015 \) |
Real period: | \( 18.05420 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.209905 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(160507\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 599 T + 160507 T^{2} )\) |
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 |
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);