Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = -2x^4 - x^3 + 5x^2 + 8x + 4$ | (homogenize, simplify) |
$y^2 + x^3y = -2x^4z^2 - x^3z^3 + 5x^2z^4 + 8xz^5 + 4z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 8x^4 - 4x^3 + 20x^2 + 32x + 16$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 8, 5, -1, -2]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 8, 5, -1, -2], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([16, 32, 20, -4, -8, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(13648\) | \(=\) | \( 2^{4} \cdot 853 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(873472\) | \(=\) | \( 2^{10} \cdot 853 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(74\) | \(=\) | \( 2 \cdot 37 \) |
\( I_4 \) | \(=\) | \(-68\) | \(=\) | \( - 2^{2} \cdot 17 \) |
\( I_6 \) | \(=\) | \(11019\) | \(=\) | \( 3 \cdot 3673 \) |
\( I_{10} \) | \(=\) | \(-3412\) | \(=\) | \( - 2^{2} \cdot 853 \) |
\( J_2 \) | \(=\) | \(148\) | \(=\) | \( 2^{2} \cdot 37 \) |
\( J_4 \) | \(=\) | \(1094\) | \(=\) | \( 2 \cdot 547 \) |
\( J_6 \) | \(=\) | \(-78308\) | \(=\) | \( - 2^{2} \cdot 19577 \) |
\( J_8 \) | \(=\) | \(-3196605\) | \(=\) | \( - 3 \cdot 5 \cdot 103 \cdot 2069 \) |
\( J_{10} \) | \(=\) | \(-873472\) | \(=\) | \( - 2^{10} \cdot 853 \) |
\( g_1 \) | \(=\) | \(-69343957/853\) | ||
\( g_2 \) | \(=\) | \(-27707191/6824\) | ||
\( g_3 \) | \(=\) | \(26800913/13648\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) |
\((2 : 0 : 1)\) | \((-2 : 4 : 1)\) | \((-3 : 5 : 1)\) | \((2 : -8 : 1)\) | \((-3 : 22 : 1)\) | \((1 : 160 : 4)\) |
\((1 : -161 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) |
\((2 : 0 : 1)\) | \((-2 : 4 : 1)\) | \((-3 : 5 : 1)\) | \((2 : -8 : 1)\) | \((-3 : 22 : 1)\) | \((1 : 160 : 4)\) |
\((1 : -161 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((-2 : 0 : 1)\) | \((0 : -4 : 1)\) |
\((0 : 4 : 1)\) | \((2 : -8 : 1)\) | \((2 : 8 : 1)\) | \((-3 : -17 : 1)\) | \((-3 : 17 : 1)\) | \((1 : -321 : 4)\) |
\((1 : 321 : 4)\) |
magma: [C![-3,5,1],C![-3,22,1],C![-2,4,1],C![-1,0,1],C![-1,1,1],C![0,-2,1],C![0,2,1],C![1,-161,4],C![1,-1,0],C![1,0,0],C![1,160,4],C![2,-8,1],C![2,0,1]]; // minimal model
magma: [C![-3,-17,1],C![-3,17,1],C![-2,0,1],C![-1,-1,1],C![-1,1,1],C![0,-4,1],C![0,4,1],C![1,-321,4],C![1,-1,0],C![1,1,0],C![1,321,4],C![2,-8,1],C![2,8,1]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -2 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.111141\) | \(\infty\) |
\((-2 : 4 : 1) - (1 : 0 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 4z^3\) | \(0.105082\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -2 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.111141\) | \(\infty\) |
\((-2 : 4 : 1) - (1 : 0 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 4z^3\) | \(0.105082\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -4 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 4z^3\) | \(0.111141\) | \(\infty\) |
\((-2 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 8z^3\) | \(0.105082\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.009148 \) |
Real period: | \( 13.44148 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.491859 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(10\) | \(4\) | \(1 + 2 T + 2 T^{2}\) | |
\(853\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 20 T + 853 T^{2} )\) |
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);