Minimal equation
Minimal equation
Simplified equation
$y^2 + x^2y = x^6 - 3x^5 + 5x^3 - 3x^2 + 2x - 3$ | (homogenize, simplify) |
$y^2 + x^2zy = x^6 - 3x^5z + 5x^3z^3 - 3x^2z^4 + 2xz^5 - 3z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 12x^5 + x^4 + 20x^3 - 12x^2 + 8x - 12$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 2, -3, 5, 0, -3, 1]), R([0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 2, -3, 5, 0, -3, 1], R![0, 0, 1]);
sage: X = HyperellipticCurve(R([-12, 8, -12, 20, 1, -12, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(20736\) | \(=\) | \( 2^{8} \cdot 3^{4} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(20736,2),R![1, -1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-331776\) | \(=\) | \( - 2^{12} \cdot 3^{4} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1284\) | \(=\) | \( 2^{2} \cdot 3 \cdot 107 \) |
\( I_4 \) | \(=\) | \(-2178\) | \(=\) | \( - 2 \cdot 3^{2} \cdot 11^{2} \) |
\( I_6 \) | \(=\) | \(-988758\) | \(=\) | \( - 2 \cdot 3^{2} \cdot 163 \cdot 337 \) |
\( I_{10} \) | \(=\) | \(-41472\) | \(=\) | \( - 2^{9} \cdot 3^{4} \) |
\( J_2 \) | \(=\) | \(1284\) | \(=\) | \( 2^{2} \cdot 3 \cdot 107 \) |
\( J_4 \) | \(=\) | \(70146\) | \(=\) | \( 2 \cdot 3^{4} \cdot 433 \) |
\( J_6 \) | \(=\) | \(5261188\) | \(=\) | \( 2^{2} \cdot 1315297 \) |
\( J_8 \) | \(=\) | \(458726019\) | \(=\) | \( 3 \cdot 67 \cdot 347 \cdot 6577 \) |
\( J_{10} \) | \(=\) | \(-331776\) | \(=\) | \( - 2^{12} \cdot 3^{4} \) |
\( g_1 \) | \(=\) | \(-42076551921/4\) | ||
\( g_2 \) | \(=\) | \(-14321977713/32\) | ||
\( g_3 \) | \(=\) | \(-15058835353/576\) |
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
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (2 : -1 : 1),\, (2 : -3 : 1),\, (3 : -9 : 2)\)
magma: [C![1,-1,0],C![1,1,0],C![2,-3,1],C![2,-1,1],C![3,-9,2]]; // minimal model
magma: [C![1,-2,0],C![1,2,0],C![2,-2,1],C![2,2,1],C![3,0,2]]; // 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: \(\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -3 : 1) - (1 : 1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 5z^3\) | \(0.112729\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -3 : 1) - (1 : 1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 5z^3\) | \(0.112729\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -2 : 1) - (1 : 2 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2x^3 + x^2z + 10z^3\) | \(0.112729\) | \(\infty\) |
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - 2z^3\) | \(0\) | \(2\) |
2-torsion field: 6.0.6718464.3
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.112729 \) |
Real period: | \( 6.735253 \) |
Tamagawa product: | \( 8 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.518521 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(12\) | \(8\) | \(1 - T\) | |
\(3\) | \(4\) | \(4\) | \(1\) | \(1 + 2 T + 3 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.60.1 | yes |
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);