Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = x^5 + 2x^4 - 3x^3$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = x^5z + 2x^4z^2 - 3x^3z^3$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 8x^4 - 12x^3 + x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, -3, 2, 1]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, -3, 2, 1], R![1, 1]);
sage: X = HyperellipticCurve(R([1, 2, 1, -12, 8, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(255382\) | \(=\) | \( 2 \cdot 127691 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(510764\) | \(=\) | \( 2^{2} \cdot 127691 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(264\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \) |
\( I_4 \) | \(=\) | \(9648\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 67 \) |
\( I_6 \) | \(=\) | \(540387\) | \(=\) | \( 3^{2} \cdot 97 \cdot 619 \) |
\( I_{10} \) | \(=\) | \(2043056\) | \(=\) | \( 2^{4} \cdot 127691 \) |
\( J_2 \) | \(=\) | \(132\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \) |
\( J_4 \) | \(=\) | \(-882\) | \(=\) | \( - 2 \cdot 3^{2} \cdot 7^{2} \) |
\( J_6 \) | \(=\) | \(4241\) | \(=\) | \( 4241 \) |
\( J_8 \) | \(=\) | \(-54528\) | \(=\) | \( - 2^{8} \cdot 3 \cdot 71 \) |
\( J_{10} \) | \(=\) | \(510764\) | \(=\) | \( 2^{2} \cdot 127691 \) |
\( g_1 \) | \(=\) | \(10018660608/127691\) | ||
\( g_2 \) | \(=\) | \(-507142944/127691\) | ||
\( g_3 \) | \(=\) | \(18473796/127691\) |
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
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((1 : -2 : 1)\) | \((-3 : 0 : 1)\) | \((-3 : 2 : 1)\) | \((1 : -2 : 4)\) | \((2 : 5 : 1)\) | \((2 : -8 : 1)\) |
\((-2 : 33 : 9)\) | \((1 : -78 : 4)\) | \((4 : -93 : 9)\) | \((-2 : -600 : 9)\) | \((4 : -960 : 9)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((1 : -2 : 1)\) | \((-3 : 0 : 1)\) | \((-3 : 2 : 1)\) | \((1 : -2 : 4)\) | \((2 : 5 : 1)\) | \((2 : -8 : 1)\) |
\((-2 : 33 : 9)\) | \((1 : -78 : 4)\) | \((4 : -93 : 9)\) | \((-2 : -600 : 9)\) | \((4 : -960 : 9)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((-3 : -2 : 1)\) |
\((-3 : 2 : 1)\) | \((-1 : -4 : 1)\) | \((-1 : 4 : 1)\) | \((2 : -13 : 1)\) | \((2 : 13 : 1)\) | \((1 : -76 : 4)\) |
\((1 : 76 : 4)\) | \((-2 : -633 : 9)\) | \((-2 : 633 : 9)\) | \((4 : -867 : 9)\) | \((4 : 867 : 9)\) |
magma: [C![-3,0,1],C![-3,2,1],C![-2,-600,9],C![-2,33,9],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-78,4],C![1,-2,1],C![1,-2,4],C![1,0,0],C![1,0,1],C![2,-8,1],C![2,5,1],C![4,-960,9],C![4,-93,9]]; // minimal model
magma: [C![-3,-2,1],C![-3,2,1],C![-2,-633,9],C![-2,633,9],C![-1,-4,1],C![-1,4,1],C![0,-1,1],C![0,1,1],C![1,-76,4],C![1,-2,1],C![1,76,4],C![1,0,0],C![1,2,1],C![2,-13,1],C![2,13,1],C![4,-867,9],C![4,867,9]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.703510\) | \(\infty\) |
\((1 : -2 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.395739\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.189784\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.703510\) | \(\infty\) |
\((1 : -2 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.395739\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.189784\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -4 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 3z^3\) | \(0.703510\) | \(\infty\) |
\((1 : -2 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 3z^3\) | \(0.395739\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.189784\) | \(\infty\) |
2-torsion field: 5.1.2043056.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.052257 \) |
Real period: | \( 16.67760 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.743071 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(127691\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 544 T + 127691 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.6.1 | no |
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);