Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + 1)y = -2x^3 - 3x^2 + x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + z^3)y = -2x^3z^3 - 3x^2z^4 + xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + x^4 - 6x^3 - 10x^2 + 4x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, -3, -2]), R([1, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, -3, -2], R![1, 0, 1, 1]);
sage: X = HyperellipticCurve(R([9, 4, -10, -6, 1, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(5501\) | \(=\) | \( 5501 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(5501\) | \(=\) | \( 5501 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(732\) | \(=\) | \( 2^{2} \cdot 3 \cdot 61 \) |
\( I_4 \) | \(=\) | \(22041\) | \(=\) | \( 3^{2} \cdot 31 \cdot 79 \) |
\( I_6 \) | \(=\) | \(6292467\) | \(=\) | \( 3^{2} \cdot 743 \cdot 941 \) |
\( I_{10} \) | \(=\) | \(-704128\) | \(=\) | \( - 2^{7} \cdot 5501 \) |
\( J_2 \) | \(=\) | \(183\) | \(=\) | \( 3 \cdot 61 \) |
\( J_4 \) | \(=\) | \(477\) | \(=\) | \( 3^{2} \cdot 53 \) |
\( J_6 \) | \(=\) | \(-26525\) | \(=\) | \( - 5^{2} \cdot 1061 \) |
\( J_8 \) | \(=\) | \(-1270401\) | \(=\) | \( - 3 \cdot 11 \cdot 137 \cdot 281 \) |
\( J_{10} \) | \(=\) | \(-5501\) | \(=\) | \( -5501 \) |
\( g_1 \) | \(=\) | \(-205236901143/5501\) | ||
\( g_2 \) | \(=\) | \(-2923288299/5501\) | ||
\( g_3 \) | \(=\) | \(888295725/5501\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
\((0 : -2 : 1)\) | \((-2 : -1 : 1)\) | \((1 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((-13 : -128 : 12)\) | \((-13 : -1431 : 12)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
\((0 : -2 : 1)\) | \((-2 : -1 : 1)\) | \((1 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((-13 : -128 : 12)\) | \((-13 : -1431 : 12)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((0 : -3 : 1)\) | \((0 : 3 : 1)\) | \((-2 : -5 : 1)\) | \((-2 : 5 : 1)\) | \((-13 : -1303 : 12)\) | \((-13 : 1303 : 12)\) |
magma: [C![-13,-1431,12],C![-13,-128,12],C![-2,-1,1],C![-2,4,1],C![-1,-1,1],C![-1,0,1],C![0,-2,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
magma: [C![-13,-1303,12],C![-13,1303,12],C![-2,-5,1],C![-2,5,1],C![-1,-1,1],C![-1,1,1],C![0,-3,1],C![0,3,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.176934\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.057232\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.176934\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.057232\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - z^3\) | \(0.176934\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - 2xz^2 - z^3\) | \(0.057232\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.010011 \) |
Real period: | \( 26.16477 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.261938 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5501\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 9 T + 5501 T^{2} )\) |
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);