Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -2x^4 - x^2 - 5x + 1$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -2x^4z^2 - x^2z^4 - 5xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 6x^4 - 3x^2 - 20x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -5, -1, 0, -2]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -5, -1, 0, -2], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([4, -20, -3, 0, -6, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(11914\) | \(=\) | \( 2 \cdot 7 \cdot 23 \cdot 37 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-548044\) | \(=\) | \( - 2^{2} \cdot 7 \cdot 23^{2} \cdot 37 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(312\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \) |
\( I_4 \) | \(=\) | \(-48240\) | \(=\) | \( - 2^{4} \cdot 3^{2} \cdot 5 \cdot 67 \) |
\( I_6 \) | \(=\) | \(-2904723\) | \(=\) | \( - 3^{2} \cdot 322747 \) |
\( I_{10} \) | \(=\) | \(2192176\) | \(=\) | \( 2^{4} \cdot 7 \cdot 23^{2} \cdot 37 \) |
\( J_2 \) | \(=\) | \(156\) | \(=\) | \( 2^{2} \cdot 3 \cdot 13 \) |
\( J_4 \) | \(=\) | \(9054\) | \(=\) | \( 2 \cdot 3^{2} \cdot 503 \) |
\( J_6 \) | \(=\) | \(-16865\) | \(=\) | \( - 5 \cdot 3373 \) |
\( J_8 \) | \(=\) | \(-21151464\) | \(=\) | \( - 2^{3} \cdot 3 \cdot 881311 \) |
\( J_{10} \) | \(=\) | \(548044\) | \(=\) | \( 2^{2} \cdot 7 \cdot 23^{2} \cdot 37 \) |
\( g_1 \) | \(=\) | \(23097394944/137011\) | ||
\( g_2 \) | \(=\) | \(8593187616/137011\) | ||
\( g_3 \) | \(=\) | \(-102606660/137011\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 3 : 1)\) |
\((-2 : 5 : 1)\) | \((-3 : 8 : 2)\) | \((-3 : 31 : 2)\) | \((-22 : 2367 : 7)\) | \((-22 : 9359 : 7)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 3 : 1)\) |
\((-2 : 5 : 1)\) | \((-3 : 8 : 2)\) | \((-3 : 31 : 2)\) | \((-22 : 2367 : 7)\) | \((-22 : 9359 : 7)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-2 : 0 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) | \((-1 : -4 : 1)\) |
\((-1 : 4 : 1)\) | \((-3 : -23 : 2)\) | \((-3 : 23 : 2)\) | \((-22 : -6992 : 7)\) | \((-22 : 6992 : 7)\) |
magma: [C![-22,2367,7],C![-22,9359,7],C![-3,8,2],C![-3,31,2],C![-2,5,1],C![-1,-1,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-22,-6992,7],C![-22,6992,7],C![-3,-23,2],C![-3,23,2],C![-2,0,1],C![-1,-4,1],C![-1,4,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.472913\) | \(\infty\) |
\((-2 : 5 : 1) - (1 : 0 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.100926\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.472913\) | \(\infty\) |
\((-2 : 5 : 1) - (1 : 0 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.100926\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 2 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + 2z^3\) | \(0.472913\) | \(\infty\) |
\((-2 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 - 6z^3\) | \(0.100926\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + 2z^3\) | \(0\) | \(2\) |
2-torsion field: 6.0.7513072.3
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.047345 \) |
Real period: | \( 11.65576 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.551851 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 3 T + 7 T^{2} )\) | |
\(23\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + 4 T + 23 T^{2} )\) | |
\(37\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 37 T^{2} )\) |
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);