Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = x^5 - 8x^4 + 14x^3 + 2x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = x^5z - 8x^4z^2 + 14x^3z^3 + 2x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 31x^4 + 58x^3 + 9x^2 - 4x$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 2, 14, -8, 1]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 2, 14, -8, 1], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([0, -4, 9, 58, -31, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(762\) | \(=\) | \( 2 \cdot 3 \cdot 127 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(82296\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 127 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(12004\) | \(=\) | \( 2^{2} \cdot 3001 \) |
\( I_4 \) | \(=\) | \(205249\) | \(=\) | \( 11 \cdot 47 \cdot 397 \) |
\( I_6 \) | \(=\) | \(810020577\) | \(=\) | \( 3 \cdot 270006859 \) |
\( I_{10} \) | \(=\) | \(10533888\) | \(=\) | \( 2^{10} \cdot 3^{4} \cdot 127 \) |
\( J_2 \) | \(=\) | \(3001\) | \(=\) | \( 3001 \) |
\( J_4 \) | \(=\) | \(366698\) | \(=\) | \( 2 \cdot 183349 \) |
\( J_6 \) | \(=\) | \(58441312\) | \(=\) | \( 2^{5} \cdot 1826291 \) |
\( J_8 \) | \(=\) | \(10228738527\) | \(=\) | \( 3^{2} \cdot 7 \cdot 3089 \cdot 52561 \) |
\( J_{10} \) | \(=\) | \(82296\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 127 \) |
\( g_1 \) | \(=\) | \(243405270090015001/82296\) | ||
\( g_2 \) | \(=\) | \(4955375073324349/41148\) | ||
\( g_3 \) | \(=\) | \(65790314289164/10287\) |
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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 2 : 1),\, (1 : -4 : 1),\, (4 : -10 : 1)\)
magma: [C![0,0,1],C![1,-4,1],C![1,0,0],C![1,2,1],C![4,-10,1]]; // minimal model
magma: [C![0,0,1],C![1,-6,1],C![1,0,0],C![1,6,1],C![4,0,1]]; // simplified model
Number of rational Weierstrass points: \(3\)
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: \(\Z/{2}\Z \oplus \Z/{12}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (4 : -10 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 4z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-5xz^2\) | \(0\) | \(2\) |
\((0 : 0 : 1) + (1 : 2 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2\) | \(0\) | \(12\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (4 : -10 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 4z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-5xz^2\) | \(0\) | \(2\) |
\((0 : 0 : 1) + (1 : 2 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2\) | \(0\) | \(12\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x (x - 4z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^2z - 9xz^2\) | \(0\) | \(2\) |
\((0 : 0 : 1) + (1 : 6 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 5xz^2\) | \(0\) | \(12\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 16.73344 \) |
Tamagawa product: | \( 12 \) |
Torsion order: | \( 24 \) |
Leading coefficient: | \( 0.348613 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(4\) | \(4\) | \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) | |
\(127\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 16 T + 127 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.120.3 | yes |
\(3\) | 3.80.1 | 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);