Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = x^6 - 3x^5 + x^4 + 2x^3 + x^2 - 2x$ | (homogenize, simplify) |
$y^2 + xz^2y = x^6 - 3x^5z + x^4z^2 + 2x^3z^3 + x^2z^4 - 2xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 12x^5 + 4x^4 + 8x^3 + 5x^2 - 8x$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 1, 2, 1, -3, 1]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 1, 2, 1, -3, 1], R![0, 1]);
sage: X = HyperellipticCurve(R([0, -8, 5, 8, 4, -12, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(154006\) | \(=\) | \( 2 \cdot 77003 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-308012\) | \(=\) | \( - 2^{2} \cdot 77003 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(976\) | \(=\) | \( 2^{4} \cdot 61 \) |
\( I_4 \) | \(=\) | \(5992\) | \(=\) | \( 2^{3} \cdot 7 \cdot 107 \) |
\( I_6 \) | \(=\) | \(1214103\) | \(=\) | \( 3 \cdot 11 \cdot 36791 \) |
\( I_{10} \) | \(=\) | \(-1232048\) | \(=\) | \( - 2^{4} \cdot 77003 \) |
\( J_2 \) | \(=\) | \(488\) | \(=\) | \( 2^{3} \cdot 61 \) |
\( J_4 \) | \(=\) | \(8924\) | \(=\) | \( 2^{2} \cdot 23 \cdot 97 \) |
\( J_6 \) | \(=\) | \(269489\) | \(=\) | \( 11 \cdot 24499 \) |
\( J_8 \) | \(=\) | \(12968214\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 131 \cdot 2357 \) |
\( J_{10} \) | \(=\) | \(-308012\) | \(=\) | \( - 2^{2} \cdot 77003 \) |
\( g_1 \) | \(=\) | \(-6918932897792/77003\) | ||
\( g_2 \) | \(=\) | \(-259274040832/77003\) | ||
\( g_3 \) | \(=\) | \(-16044297104/77003\) |
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 | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) | \((2 : 0 : 1)\) |
\((-1 : -2 : 1)\) | \((2 : -2 : 1)\) | \((-1 : 3 : 1)\) | \((-2 : -12 : 1)\) | \((-2 : 14 : 1)\) | \((-1 : -19 : 3)\) |
\((-1 : 28 : 3)\) | \((8 : -68 : 5)\) | \((8 : -132 : 5)\) | \((18 : -966 : 11)\) | \((18 : -1212 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) | \((2 : 0 : 1)\) |
\((-1 : -2 : 1)\) | \((2 : -2 : 1)\) | \((-1 : 3 : 1)\) | \((-2 : -12 : 1)\) | \((-2 : 14 : 1)\) | \((-1 : -19 : 3)\) |
\((-1 : 28 : 3)\) | \((8 : -68 : 5)\) | \((8 : -132 : 5)\) | \((18 : -966 : 11)\) | \((18 : -1212 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : 0 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((2 : -2 : 1)\) |
\((2 : 2 : 1)\) | \((-1 : -5 : 1)\) | \((-1 : 5 : 1)\) | \((-2 : -26 : 1)\) | \((-2 : 26 : 1)\) | \((-1 : -47 : 3)\) |
\((-1 : 47 : 3)\) | \((8 : -64 : 5)\) | \((8 : 64 : 5)\) | \((18 : -246 : 11)\) | \((18 : 246 : 11)\) |
magma: [C![-2,-12,1],C![-2,14,1],C![-1,-19,3],C![-1,-2,1],C![-1,3,1],C![-1,28,3],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-2,1],C![2,0,1],C![8,-132,5],C![8,-68,5],C![18,-1212,11],C![18,-966,11]]; // minimal model
magma: [C![-2,-26,1],C![-2,26,1],C![-1,-47,3],C![-1,-5,1],C![-1,5,1],C![-1,47,3],C![0,0,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![2,-2,1],C![2,2,1],C![8,-64,5],C![8,64,5],C![18,-246,11],C![18,246,11]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (2 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.494631\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.368691\) | \(\infty\) |
\((0 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.338870\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (2 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.494631\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.368691\) | \(\infty\) |
\((0 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.338870\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (2 : 2 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.494631\) | \(\infty\) |
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + 2z^3\) | \(0.368691\) | \(\infty\) |
\((0 : 0 : 1) + (1 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.338870\) | \(\infty\) |
2-torsion field: 5.3.1232048.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.044393 \) |
Real period: | \( 15.80830 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.403564 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(77003\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 229 T + 77003 T^{2} )\) |
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);