Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = 2x^4 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = 2x^4z^2 + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 8x^4 + 2x^3 + 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, 0, 2]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, 0, 2], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, 4, 0, 2, 8, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(104482\) | \(=\) | \( 2 \cdot 7 \cdot 17 \cdot 439 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(208964\) | \(=\) | \( 2^{2} \cdot 7 \cdot 17 \cdot 439 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(108\) | \(=\) | \( 2^{2} \cdot 3^{3} \) |
\( I_4 \) | \(=\) | \(14937\) | \(=\) | \( 3 \cdot 13 \cdot 383 \) |
\( I_6 \) | \(=\) | \(608067\) | \(=\) | \( 3^{4} \cdot 7507 \) |
\( I_{10} \) | \(=\) | \(-26747392\) | \(=\) | \( - 2^{9} \cdot 7 \cdot 17 \cdot 439 \) |
\( J_2 \) | \(=\) | \(27\) | \(=\) | \( 3^{3} \) |
\( J_4 \) | \(=\) | \(-592\) | \(=\) | \( - 2^{4} \cdot 37 \) |
\( J_6 \) | \(=\) | \(-3732\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 311 \) |
\( J_8 \) | \(=\) | \(-112807\) | \(=\) | \( -112807 \) |
\( J_{10} \) | \(=\) | \(-208964\) | \(=\) | \( - 2^{2} \cdot 7 \cdot 17 \cdot 439 \) |
\( g_1 \) | \(=\) | \(-14348907/208964\) | ||
\( g_2 \) | \(=\) | \(2913084/52241\) | ||
\( g_3 \) | \(=\) | \(680157/52241\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : 1 : 1)\) | \((1 : -3 : 1)\) | \((-2 : -3 : 1)\) | \((3 : 5 : 1)\) | \((-2 : 10 : 1)\) | \((-1 : -31 : 4)\) |
\((-1 : -32 : 4)\) | \((3 : -33 : 1)\) | \((-5 : 45 : 6)\) | \((-5 : -136 : 6)\) | \((3 : 248 : 10)\) | \((3 : -1275 : 10)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : 1 : 1)\) | \((1 : -3 : 1)\) | \((-2 : -3 : 1)\) | \((3 : 5 : 1)\) | \((-2 : 10 : 1)\) | \((-1 : -31 : 4)\) |
\((-1 : -32 : 4)\) | \((3 : -33 : 1)\) | \((-5 : 45 : 6)\) | \((-5 : -136 : 6)\) | \((3 : 248 : 10)\) | \((3 : -1275 : 10)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((-1 : -1 : 4)\) | \((-1 : 1 : 4)\) | \((1 : -4 : 1)\) | \((1 : 4 : 1)\) | \((-2 : -13 : 1)\) | \((-2 : 13 : 1)\) |
\((3 : -38 : 1)\) | \((3 : 38 : 1)\) | \((-5 : -181 : 6)\) | \((-5 : 181 : 6)\) | \((3 : -1523 : 10)\) | \((3 : 1523 : 10)\) |
magma: [C![-5,-136,6],C![-5,45,6],C![-2,-3,1],C![-2,10,1],C![-1,-32,4],C![-1,-31,4],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![3,-1275,10],C![3,-33,1],C![3,5,1],C![3,248,10]]; // minimal model
magma: [C![-5,-181,6],C![-5,181,6],C![-2,-13,1],C![-2,13,1],C![-1,-1,4],C![-1,1,4],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,4,1],C![3,-1523,10],C![3,-38,1],C![3,38,1],C![3,1523,10]]; // simplified model
Number of rational Weierstrass points: \(0\)
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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.430857\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.262951\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.342396\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.430857\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.262951\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.342396\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 2 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 4xz^2 - z^3\) | \(0.430857\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.262951\) | \(\infty\) |
\((-1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.342396\) | \(\infty\) |
2-torsion field: 6.2.3343424.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.030275 \) |
Real period: | \( 14.42719 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.873572 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 7 T^{2} )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 6 T + 17 T^{2} )\) | |
\(439\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 24 T + 439 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .
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);