Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x)y = -2x^4 - x + 1$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2)y = -2x^4z^2 - xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 - 5x^4 + 2x^3 + x^2 - 4x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -1, 0, 0, -2]), R([0, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -1, 0, 0, -2], R![0, 1, 1, 1]);
sage: X = HyperellipticCurve(R([4, -4, 1, 2, -5, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(52498\) | \(=\) | \( 2 \cdot 26249 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(104996\) | \(=\) | \( 2^{2} \cdot 26249 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(588\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( I_4 \) | \(=\) | \(6873\) | \(=\) | \( 3 \cdot 29 \cdot 79 \) |
\( I_6 \) | \(=\) | \(1630563\) | \(=\) | \( 3 \cdot 11 \cdot 49411 \) |
\( I_{10} \) | \(=\) | \(-13439488\) | \(=\) | \( - 2^{9} \cdot 26249 \) |
\( J_2 \) | \(=\) | \(147\) | \(=\) | \( 3 \cdot 7^{2} \) |
\( J_4 \) | \(=\) | \(614\) | \(=\) | \( 2 \cdot 307 \) |
\( J_6 \) | \(=\) | \(-3600\) | \(=\) | \( - 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
\( J_8 \) | \(=\) | \(-226549\) | \(=\) | \( -226549 \) |
\( J_{10} \) | \(=\) | \(-104996\) | \(=\) | \( - 2^{2} \cdot 26249 \) |
\( g_1 \) | \(=\) | \(-68641485507/104996\) | ||
\( g_2 \) | \(=\) | \(-975192561/52498\) | ||
\( g_3 \) | \(=\) | \(19448100/26249\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : -2 : 1)\) | \((2 : -3 : 1)\) | \((-1 : -8 : 2)\) | \((-1 : 11 : 2)\) | \((2 : -11 : 1)\) |
\((-4 : 13 : 1)\) | \((3 : -17 : 2)\) | \((-4 : 39 : 1)\) | \((3 : -40 : 2)\) | \((4 : -49 : 3)\) | \((4 : -99 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : -2 : 1)\) | \((2 : -3 : 1)\) | \((-1 : -8 : 2)\) | \((-1 : 11 : 2)\) | \((2 : -11 : 1)\) |
\((-4 : 13 : 1)\) | \((3 : -17 : 2)\) | \((-4 : 39 : 1)\) | \((3 : -40 : 2)\) | \((4 : -49 : 3)\) | \((4 : -99 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((0 : -2 : 1)\) | \((0 : 2 : 1)\) | \((2 : -8 : 1)\) | \((2 : 8 : 1)\) | \((-1 : -19 : 2)\) | \((-1 : 19 : 2)\) |
\((3 : -23 : 2)\) | \((3 : 23 : 2)\) | \((-4 : -26 : 1)\) | \((-4 : 26 : 1)\) | \((4 : -50 : 3)\) | \((4 : 50 : 3)\) |
magma: [C![-4,13,1],C![-4,39,1],C![-1,-8,2],C![-1,0,1],C![-1,1,1],C![-1,11,2],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-11,1],C![2,-3,1],C![3,-40,2],C![3,-17,2],C![4,-99,3],C![4,-49,3]]; // minimal model
magma: [C![-4,-26,1],C![-4,26,1],C![-1,-19,2],C![-1,-1,1],C![-1,1,1],C![-1,19,2],C![0,-2,1],C![0,2,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![2,-8,1],C![2,8,1],C![3,-23,2],C![3,23,2],C![4,-50,3],C![4,50,3]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.262486\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.363696\) | \(\infty\) |
\((0 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0.256046\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.262486\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.363696\) | \(\infty\) |
\((0 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0.256046\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -2 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + x^2z + xz^2 - 2z^3\) | \(0.262486\) | \(\infty\) |
\((0 : -2 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 - 2z^3\) | \(0.363696\) | \(\infty\) |
\((0 : 2 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - 3xz^2 + 2z^3\) | \(0.256046\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.020395 \) |
Real period: | \( 16.21731 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.661515 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(26249\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 167 T + 26249 T^{2} )\) |
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);