Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^5 + 2x^4 - x^2 - 2x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^5z + 2x^4z^2 - x^2z^4 - 2xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 10x^4 + 2x^3 - 3x^2 - 6x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, -1, 0, 2, 1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, -1, 0, 2, 1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, -6, -3, 2, 10, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(17364\) | \(=\) | \( 2^{2} \cdot 3 \cdot 1447 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(937656\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 1447 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(348\) | \(=\) | \( 2^{2} \cdot 3 \cdot 29 \) |
\( I_4 \) | \(=\) | \(30225\) | \(=\) | \( 3 \cdot 5^{2} \cdot 13 \cdot 31 \) |
\( I_6 \) | \(=\) | \(3242247\) | \(=\) | \( 3 \cdot 1080749 \) |
\( I_{10} \) | \(=\) | \(-120019968\) | \(=\) | \( - 2^{10} \cdot 3^{4} \cdot 1447 \) |
\( J_2 \) | \(=\) | \(87\) | \(=\) | \( 3 \cdot 29 \) |
\( J_4 \) | \(=\) | \(-944\) | \(=\) | \( - 2^{4} \cdot 59 \) |
\( J_6 \) | \(=\) | \(-13072\) | \(=\) | \( - 2^{4} \cdot 19 \cdot 43 \) |
\( J_8 \) | \(=\) | \(-507100\) | \(=\) | \( - 2^{2} \cdot 5^{2} \cdot 11 \cdot 461 \) |
\( J_{10} \) | \(=\) | \(-937656\) | \(=\) | \( - 2^{3} \cdot 3^{4} \cdot 1447 \) |
\( g_1 \) | \(=\) | \(-61533447/11576\) | ||
\( g_2 \) | \(=\) | \(2877902/4341\) | ||
\( g_3 \) | \(=\) | \(1374194/13023\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -1 : 1)\) |
\((-2 : 0 : 1)\) | \((-1 : 2 : 1)\) | \((1 : -3 : 1)\) | \((-1 : 6 : 2)\) | \((-2 : 9 : 1)\) | \((-1 : -9 : 2)\) |
\((9 : 323 : 4)\) | \((9 : -1260 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -1 : 1)\) |
\((-2 : 0 : 1)\) | \((-1 : 2 : 1)\) | \((1 : -3 : 1)\) | \((-1 : 6 : 2)\) | \((-2 : 9 : 1)\) | \((-1 : -9 : 2)\) |
\((9 : 323 : 4)\) | \((9 : -1260 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) |
\((1 : -3 : 1)\) | \((1 : 3 : 1)\) | \((-2 : -9 : 1)\) | \((-2 : 9 : 1)\) | \((-1 : -15 : 2)\) | \((-1 : 15 : 2)\) |
\((9 : -1583 : 4)\) | \((9 : 1583 : 4)\) |
magma: [C![-2,0,1],C![-2,9,1],C![-1,-9,2],C![-1,-1,1],C![-1,2,1],C![-1,6,2],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![9,-1260,4],C![9,323,4]]; // minimal model
magma: [C![-2,-9,1],C![-2,9,1],C![-1,-15,2],C![-1,-3,1],C![-1,3,1],C![-1,15,2],C![0,-1,1],C![0,1,1],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![9,-1583,4],C![9,1583,4]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z\) | \(0.455308\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - 2z^3\) | \(0.010337\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z\) | \(0.455308\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - 2z^3\) | \(0.010337\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 2x^2z + xz^2 + z^3\) | \(0.455308\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2 - 3z^3\) | \(0.010337\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.004696 \) |
Real period: | \( 12.51432 \) |
Tamagawa product: | \( 12 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.705311 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(3\) | \(3\) | \(1 + T + T^{2}\) | |
\(3\) | \(1\) | \(4\) | \(4\) | \(( 1 - T )( 1 + 3 T + 3 T^{2} )\) | |
\(1447\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 18 T + 1447 T^{2} )\) |
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);