Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^5 + x^4 + 5x^2 + 11x + 6$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^5z + x^4z^2 + 5x^2z^4 + 11xz^5 + 6z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 4x^4 + 2x^3 + 20x^2 + 44x + 25$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([6, 11, 5, 0, 1, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![6, 11, 5, 0, 1, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([25, 44, 20, 2, 4, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(231094\) | \(=\) | \( 2 \cdot 115547 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(462188\) | \(=\) | \( 2^{2} \cdot 115547 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(108\) | \(=\) | \( 2^{2} \cdot 3^{3} \) |
\( I_4 \) | \(=\) | \(-10503\) | \(=\) | \( - 3^{3} \cdot 389 \) |
\( I_6 \) | \(=\) | \(604755\) | \(=\) | \( 3^{2} \cdot 5 \cdot 89 \cdot 151 \) |
\( I_{10} \) | \(=\) | \(-59160064\) | \(=\) | \( - 2^{9} \cdot 115547 \) |
\( J_2 \) | \(=\) | \(27\) | \(=\) | \( 3^{3} \) |
\( J_4 \) | \(=\) | \(468\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 13 \) |
\( J_6 \) | \(=\) | \(-11636\) | \(=\) | \( - 2^{2} \cdot 2909 \) |
\( J_8 \) | \(=\) | \(-133299\) | \(=\) | \( - 3^{3} \cdot 4937 \) |
\( J_{10} \) | \(=\) | \(-462188\) | \(=\) | \( - 2^{2} \cdot 115547 \) |
\( g_1 \) | \(=\) | \(-14348907/462188\) | ||
\( g_2 \) | \(=\) | \(-2302911/115547\) | ||
\( g_3 \) | \(=\) | \(2120661/115547\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 2 : 1)\) | \((0 : -3 : 1)\) | \((-2 : 3 : 1)\) |
\((1 : 4 : 1)\) | \((-2 : 4 : 1)\) | \((1 : -6 : 1)\) | \((-3 : 8 : 1)\) | \((-3 : 18 : 1)\) | \((-3 : 36 : 4)\) |
\((-3 : -73 : 4)\) | \((-19 : 2874 : 10)\) | \((-19 : 2985 : 10)\) | \((-29 : 5855 : 30)\) | \((-29 : -8466 : 30)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 2 : 1)\) | \((0 : -3 : 1)\) | \((-2 : 3 : 1)\) |
\((1 : 4 : 1)\) | \((-2 : 4 : 1)\) | \((1 : -6 : 1)\) | \((-3 : 8 : 1)\) | \((-3 : 18 : 1)\) | \((-3 : 36 : 4)\) |
\((-3 : -73 : 4)\) | \((-19 : 2874 : 10)\) | \((-19 : 2985 : 10)\) | \((-29 : 5855 : 30)\) | \((-29 : -8466 : 30)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : 0 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) | \((0 : -5 : 1)\) |
\((0 : 5 : 1)\) | \((1 : -10 : 1)\) | \((1 : 10 : 1)\) | \((-3 : -10 : 1)\) | \((-3 : 10 : 1)\) | \((-3 : -109 : 4)\) |
\((-3 : 109 : 4)\) | \((-19 : -111 : 10)\) | \((-19 : 111 : 10)\) | \((-29 : -14321 : 30)\) | \((-29 : 14321 : 30)\) |
magma: [C![-29,-8466,30],C![-29,5855,30],C![-19,2874,10],C![-19,2985,10],C![-3,-73,4],C![-3,8,1],C![-3,18,1],C![-3,36,4],C![-2,3,1],C![-2,4,1],C![-1,0,1],C![0,-3,1],C![0,2,1],C![1,-6,1],C![1,-1,0],C![1,0,0],C![1,4,1]]; // minimal model
magma: [C![-29,-14321,30],C![-29,14321,30],C![-19,-111,10],C![-19,111,10],C![-3,-109,4],C![-3,-10,1],C![-3,10,1],C![-3,109,4],C![-2,-1,1],C![-2,1,1],C![-1,0,1],C![0,-5,1],C![0,5,1],C![1,-10,1],C![1,-1,0],C![1,1,0],C![1,10,1]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 3 : 1) + (1 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - 3z^3\) | \(0.882726\) | \(\infty\) |
\((-2 : 4 : 1) - (1 : 0 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 4z^3\) | \(0.471733\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.141521\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 3 : 1) + (1 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - 3z^3\) | \(0.882726\) | \(\infty\) |
\((-2 : 4 : 1) - (1 : 0 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 4z^3\) | \(0.471733\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.141521\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : -1 : 1) + (1 : -10 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 6xz^2 - 5z^3\) | \(0.882726\) | \(\infty\) |
\((-2 : 1 : 1) - (1 : 1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 7z^3\) | \(0.471733\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.141521\) | \(\infty\) |
2-torsion field: 5.1.1848752.2
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.054775 \) |
Real period: | \( 12.24822 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.341800 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(115547\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 628 T + 115547 T^{2} )\) |
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);