Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^5 - 4x^4 + 4x^2 - 2x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^5z - 4x^4z^2 + 4x^2z^4 - 2xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 - 16x^4 + 2x^3 + 16x^2 - 8x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 4, 0, -4, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 4, 0, -4, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, -8, 16, 2, -16, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(104294\) | \(=\) | \( 2 \cdot 52147 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(208588\) | \(=\) | \( 2^{2} \cdot 52147 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1300\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 13 \) |
\( I_4 \) | \(=\) | \(55225\) | \(=\) | \( 5^{2} \cdot 47^{2} \) |
\( I_6 \) | \(=\) | \(20649037\) | \(=\) | \( 20649037 \) |
\( I_{10} \) | \(=\) | \(26699264\) | \(=\) | \( 2^{9} \cdot 52147 \) |
\( J_2 \) | \(=\) | \(325\) | \(=\) | \( 5^{2} \cdot 13 \) |
\( J_4 \) | \(=\) | \(2100\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \) |
\( J_6 \) | \(=\) | \(404\) | \(=\) | \( 2^{2} \cdot 101 \) |
\( J_8 \) | \(=\) | \(-1069675\) | \(=\) | \( - 5^{2} \cdot 42787 \) |
\( J_{10} \) | \(=\) | \(208588\) | \(=\) | \( 2^{2} \cdot 52147 \) |
\( g_1 \) | \(=\) | \(3625908203125/208588\) | ||
\( g_2 \) | \(=\) | \(18022265625/52147\) | ||
\( g_3 \) | \(=\) | \(10668125/52147\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : -2 : 2)\) | \((2 : -4 : 1)\) | \((2 : -5 : 1)\) | \((1 : -7 : 2)\) | \((1 : -13 : 3)\) |
\((1 : -15 : 3)\) | \((3 : -15 : 4)\) | \((3 : -76 : 4)\) | \((42 : -13908 : 17)\) | \((42 : -65093 : 17)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : -2 : 2)\) | \((2 : -4 : 1)\) | \((2 : -5 : 1)\) | \((1 : -7 : 2)\) | \((1 : -13 : 3)\) |
\((1 : -15 : 3)\) | \((3 : -15 : 4)\) | \((3 : -76 : 4)\) | \((42 : -13908 : 17)\) | \((42 : -65093 : 17)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((2 : -1 : 1)\) | \((2 : 1 : 1)\) | \((1 : -2 : 3)\) | \((1 : 2 : 3)\) | \((1 : -5 : 2)\) |
\((1 : 5 : 2)\) | \((3 : -61 : 4)\) | \((3 : 61 : 4)\) | \((42 : -51185 : 17)\) | \((42 : 51185 : 17)\) |
magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-15,3],C![1,-13,3],C![1,-7,2],C![1,-2,2],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-5,1],C![2,-4,1],C![3,-76,4],C![3,-15,4],C![42,-65093,17],C![42,-13908,17]]; // minimal model
magma: [C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-2,3],C![1,2,3],C![1,-5,2],C![1,5,2],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-1,1],C![2,1,1],C![3,-61,4],C![3,61,4],C![42,-51185,17],C![42,51185,17]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.829863\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.311314\) | \(\infty\) |
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.109942\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.829863\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.311314\) | \(\infty\) |
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.109942\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.829863\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.311314\) | \(\infty\) |
\((0 : -1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.109942\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.025629 \) |
Real period: | \( 18.79689 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.963526 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(52147\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 318 T + 52147 T^{2} )\) |
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);