Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = x^5 + 12x^4 - 7x^3 - 3x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = x^5z + 12x^4z^2 - 7x^3z^3 - 3x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 49x^4 - 28x^3 - 10x^2 + 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -3, -7, 12, 1]), R([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -3, -7, 12, 1], R![1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 4, -10, -28, 49, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1197\) | \(=\) | \( 3^{2} \cdot 7 \cdot 19 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-410571\) | \(=\) | \( - 3^{2} \cdot 7^{4} \cdot 19 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(3296\) | \(=\) | \( 2^{5} \cdot 103 \) |
\( I_4 \) | \(=\) | \(706780\) | \(=\) | \( 2^{2} \cdot 5 \cdot 35339 \) |
\( I_6 \) | \(=\) | \(578353015\) | \(=\) | \( 5 \cdot 8699 \cdot 13297 \) |
\( I_{10} \) | \(=\) | \(-1642284\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 19 \) |
\( J_2 \) | \(=\) | \(1648\) | \(=\) | \( 2^{4} \cdot 103 \) |
\( J_4 \) | \(=\) | \(-4634\) | \(=\) | \( - 2 \cdot 7 \cdot 331 \) |
\( J_6 \) | \(=\) | \(23921\) | \(=\) | \( 19 \cdot 1259 \) |
\( J_8 \) | \(=\) | \(4486963\) | \(=\) | \( 13 \cdot 17 \cdot 79 \cdot 257 \) |
\( J_{10} \) | \(=\) | \(-410571\) | \(=\) | \( - 3^{2} \cdot 7^{4} \cdot 19 \) |
\( g_1 \) | \(=\) | \(-12155869717331968/410571\) | ||
\( g_2 \) | \(=\) | \(2962986082304/58653\) | ||
\( g_3 \) | \(=\) | \(-3419323136/21609\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -34 : 4)\)
magma: [C![-1,-34,4],C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model
magma: [C![-1,0,4],C![0,-1,1],C![0,1,1],C![1,0,0]]; // simplified model
Number of rational Weierstrass points: \(2\)
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: \(\Z/{10}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -34 : 4) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (4x + z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(17xz^2\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -34 : 4) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (4x + z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(17xz^2\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x (4x + z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(x^2z + 34xz^2 + z^3\) | \(0\) | \(10\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 18.77804 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 10 \) |
Leading coefficient: | \( 0.375560 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(2\) | \(2\) | \(1\) | \(1 + T^{2}\) | |
\(7\) | \(1\) | \(4\) | \(2\) | \(( 1 + T )( 1 - 3 T + 7 T^{2} )\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 19 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.30.3 | yes |
\(5\) | not computed | 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);