Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x + 1)y = x^5 - 4x^3 - x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^5z - 4x^3z^3 - x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 6x^5 + 3x^4 - 12x^3 - x^2 + 6x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -1, -4, 0, 1]), R([1, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -1, -4, 0, 1], R![1, 1, 1, 1]);
sage: X = HyperellipticCurve(R([1, 6, -1, -12, 3, 6, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(95332\) | \(=\) | \( 2^{2} \cdot 23833 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-381328\) | \(=\) | \( - 2^{4} \cdot 23833 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(264\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \) |
\( I_4 \) | \(=\) | \(2661\) | \(=\) | \( 3 \cdot 887 \) |
\( I_6 \) | \(=\) | \(188655\) | \(=\) | \( 3 \cdot 5 \cdot 12577 \) |
\( I_{10} \) | \(=\) | \(-47666\) | \(=\) | \( - 2 \cdot 23833 \) |
\( J_2 \) | \(=\) | \(264\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \) |
\( J_4 \) | \(=\) | \(1130\) | \(=\) | \( 2 \cdot 5 \cdot 113 \) |
\( J_6 \) | \(=\) | \(4992\) | \(=\) | \( 2^{7} \cdot 3 \cdot 13 \) |
\( J_8 \) | \(=\) | \(10247\) | \(=\) | \( 10247 \) |
\( J_{10} \) | \(=\) | \(-381328\) | \(=\) | \( - 2^{4} \cdot 23833 \) |
\( g_1 \) | \(=\) | \(-80149284864/23833\) | ||
\( g_2 \) | \(=\) | \(-1299481920/23833\) | ||
\( g_3 \) | \(=\) | \(-21745152/23833\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : -1 : 2)\) | \((-2 : 2 : 1)\) | \((1 : -3 : 1)\) | \((-2 : 3 : 1)\) | \((3 : 3 : 1)\) |
\((-2 : -6 : 3)\) | \((-2 : -7 : 3)\) | \((1 : -14 : 2)\) | \((3 : -43 : 1)\) | \((-5 : 45 : 1)\) | \((-5 : 59 : 1)\) |
\((-1 : -74 : 6)\) | \((-1 : -111 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : -1 : 2)\) | \((-2 : 2 : 1)\) | \((1 : -3 : 1)\) | \((-2 : 3 : 1)\) | \((3 : 3 : 1)\) |
\((-2 : -6 : 3)\) | \((-2 : -7 : 3)\) | \((1 : -14 : 2)\) | \((3 : -43 : 1)\) | \((-5 : 45 : 1)\) | \((-5 : 59 : 1)\) |
\((-1 : -74 : 6)\) | \((-1 : -111 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) |
\((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((-2 : -1 : 3)\) | \((-2 : 1 : 3)\) |
\((1 : -13 : 2)\) | \((1 : 13 : 2)\) | \((-5 : -14 : 1)\) | \((-5 : 14 : 1)\) | \((-1 : -37 : 6)\) | \((-1 : 37 : 6)\) |
\((3 : -46 : 1)\) | \((3 : 46 : 1)\) |
magma: [C![-5,45,1],C![-5,59,1],C![-2,-7,3],C![-2,-6,3],C![-2,2,1],C![-2,3,1],C![-1,-111,6],C![-1,-74,6],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-14,2],C![1,-3,1],C![1,-1,0],C![1,-1,1],C![1,-1,2],C![1,0,0],C![3,-43,1],C![3,3,1]]; // minimal model
magma: [C![-5,-14,1],C![-5,14,1],C![-2,-1,3],C![-2,1,3],C![-2,-1,1],C![-2,1,1],C![-1,-37,6],C![-1,37,6],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-13,2],C![1,-2,1],C![1,-1,0],C![1,2,1],C![1,13,2],C![1,1,0],C![3,-46,1],C![3,46,1]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.322349\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.317001\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.188380\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.322349\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.317001\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.188380\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + 3xz^2 + z^3\) | \(0.322349\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 - z^3\) | \(0.317001\) | \(\infty\) |
\((-1 : -2 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + x^2z + xz^2 - 3z^3\) | \(0.188380\) | \(\infty\) |
2-torsion field: 6.4.6101248.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.016698 \) |
Real period: | \( 17.49582 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.876445 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(4\) | \(3\) | \(1 + 2 T + 2 T^{2}\) | |
\(23833\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 70 T + 23833 T^{2} )\) |
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);