Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x + 1)y = 4x^3 + 5x^2 - 3x$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = 4x^3z^3 + 5x^2z^4 - 3xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 3x^4 + 20x^3 + 23x^2 - 10x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 5, 4]), R([1, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 5, 4], R![1, 1, 1, 1]);
sage: X = HyperellipticCurve(R([1, -10, 23, 20, 3, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(101291\) | \(=\) | \( 199 \cdot 509 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-101291\) | \(=\) | \( - 199 \cdot 509 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(64\) | \(=\) | \( 2^{6} \) |
\( I_4 \) | \(=\) | \(67360\) | \(=\) | \( 2^{5} \cdot 5 \cdot 421 \) |
\( I_6 \) | \(=\) | \(-4243856\) | \(=\) | \( - 2^{4} \cdot 265241 \) |
\( I_{10} \) | \(=\) | \(-405164\) | \(=\) | \( - 2^{2} \cdot 199 \cdot 509 \) |
\( J_2 \) | \(=\) | \(32\) | \(=\) | \( 2^{5} \) |
\( J_4 \) | \(=\) | \(-11184\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 233 \) |
\( J_6 \) | \(=\) | \(571408\) | \(=\) | \( 2^{4} \cdot 71 \cdot 503 \) |
\( J_8 \) | \(=\) | \(-26699200\) | \(=\) | \( - 2^{6} \cdot 5^{2} \cdot 11 \cdot 37 \cdot 41 \) |
\( J_{10} \) | \(=\) | \(-101291\) | \(=\) | \( - 199 \cdot 509 \) |
\( g_1 \) | \(=\) | \(-33554432/101291\) | ||
\( g_2 \) | \(=\) | \(366477312/101291\) | ||
\( g_3 \) | \(=\) | \(-585121792/101291\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((1 : 1 : 2)\) | \((-2 : 2 : 1)\) | \((-2 : 3 : 1)\) | \((1 : -16 : 2)\) | \((3 : 16 : 2)\) | \((-2 : 41 : 3)\) |
\((-2 : -54 : 3)\) | \((3 : -81 : 2)\) | \((-21 : 2576 : 10)\) | \((-21 : 3375 : 10)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((1 : 1 : 2)\) | \((-2 : 2 : 1)\) | \((-2 : 3 : 1)\) | \((1 : -16 : 2)\) | \((3 : 16 : 2)\) | \((-2 : 41 : 3)\) |
\((-2 : -54 : 3)\) | \((3 : -81 : 2)\) | \((-21 : 2576 : 10)\) | \((-21 : 3375 : 10)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) |
\((-1 : -4 : 1)\) | \((-1 : 4 : 1)\) | \((1 : -17 : 2)\) | \((1 : 17 : 2)\) | \((-2 : -95 : 3)\) | \((-2 : 95 : 3)\) |
\((3 : -97 : 2)\) | \((3 : 97 : 2)\) | \((-21 : -799 : 10)\) | \((-21 : 799 : 10)\) |
magma: [C![-21,2576,10],C![-21,3375,10],C![-2,-54,3],C![-2,2,1],C![-2,3,1],C![-2,41,3],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-16,2],C![1,-1,0],C![1,0,0],C![1,1,2],C![3,-81,2],C![3,16,2]]; // minimal model
magma: [C![-21,-799,10],C![-21,799,10],C![-2,-95,3],C![-2,-1,1],C![-2,1,1],C![-2,95,3],C![-1,-4,1],C![-1,4,1],C![0,-1,1],C![0,1,1],C![1,-17,2],C![1,-1,0],C![1,1,0],C![1,17,2],C![3,-97,2],C![3,97,2]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.494311\) | \(\infty\) |
\((-2 : 2 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.260271\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.341158\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.494311\) | \(\infty\) |
\((-2 : 2 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.260271\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.341158\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 - z^3\) | \(0.494311\) | \(\infty\) |
\((-2 : -1 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - xz^2 + z^3\) | \(0.260271\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + x^2z + xz^2 - z^3\) | \(0.341158\) | \(\infty\) |
2-torsion field: 6.0.1620656.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.039401 \) |
Real period: | \( 20.81300 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.820069 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(199\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 5 T + 199 T^{2} )\) | |
\(509\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 30 T + 509 T^{2} )\) |
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);