Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + 1)y = 2x^4 - 2x^2 - 7x + 5$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + z^3)y = 2x^4z^2 - 2x^2z^4 - 7xz^5 + 5z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 9x^4 + 2x^3 - 6x^2 - 28x + 21$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, -7, -2, 0, 2]), R([1, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, -7, -2, 0, 2], R![1, 0, 1, 1]);
sage: X = HyperellipticCurve(R([21, -28, -6, 2, 9, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(17689\) | \(=\) | \( 7^{2} \cdot 19^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(866761\) | \(=\) | \( 7^{4} \cdot 19^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(3196\) | \(=\) | \( 2^{2} \cdot 17 \cdot 47 \) |
\( I_4 \) | \(=\) | \(1064809\) | \(=\) | \( 43 \cdot 24763 \) |
\( I_6 \) | \(=\) | \(806830683\) | \(=\) | \( 3 \cdot 67 \cdot 257 \cdot 15619 \) |
\( I_{10} \) | \(=\) | \(-110945408\) | \(=\) | \( - 2^{7} \cdot 7^{4} \cdot 19^{2} \) |
\( J_2 \) | \(=\) | \(799\) | \(=\) | \( 17 \cdot 47 \) |
\( J_4 \) | \(=\) | \(-17767\) | \(=\) | \( - 109 \cdot 163 \) |
\( J_6 \) | \(=\) | \(-178217\) | \(=\) | \( - 13 \cdot 13709 \) |
\( J_8 \) | \(=\) | \(-114515418\) | \(=\) | \( - 2 \cdot 3 \cdot 19085903 \) |
\( J_{10} \) | \(=\) | \(-866761\) | \(=\) | \( - 7^{4} \cdot 19^{2} \) |
\( g_1 \) | \(=\) | \(-325637113603999/866761\) | ||
\( g_2 \) | \(=\) | \(9062633983033/866761\) | ||
\( g_3 \) | \(=\) | \(113773911017/866761\) |
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);
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Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((1 : -1 : 1)\) | \((1 : -2 : 1)\) | \((-1 : 3 : 1)\) | \((-1 : -4 : 1)\) |
\((5 : 17 : 2)\) | \((3 : -32 : 4)\) | \((3 : -95 : 4)\) | \((5 : -200 : 2)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((1 : -1 : 1)\) | \((1 : -2 : 1)\) | \((-1 : 3 : 1)\) | \((-1 : -4 : 1)\) |
\((5 : 17 : 2)\) | \((3 : -32 : 4)\) | \((3 : -95 : 4)\) | \((5 : -200 : 2)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((-1 : -7 : 1)\) | \((-1 : 7 : 1)\) |
\((3 : -63 : 4)\) | \((3 : 63 : 4)\) | \((5 : -217 : 2)\) | \((5 : 217 : 2)\) |
magma: [C![-1,-4,1],C![-1,3,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![3,-95,4],C![3,-32,4],C![5,-200,2],C![5,17,2]]; // minimal model
magma: [C![-1,-7,1],C![-1,7,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![3,-63,4],C![3,63,4],C![5,-217,2],C![5,217,2]]; // simplified model
Number of rational Weierstrass points: \(0\)
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: \(\Z \oplus \Z \oplus \Z/{9}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((5 : -200 : 2) - (1 : 0 : 0)\) | \(z (2x - 5z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(-8x^3 - 75z^3\) | \(2.254858\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.713662\) | \(\infty\) |
\((-1 : -4 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4z^3\) | \(0\) | \(9\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((5 : -200 : 2) - (1 : 0 : 0)\) | \(z (2x - 5z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(-8x^3 - 75z^3\) | \(2.254858\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.713662\) | \(\infty\) |
\((-1 : -4 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4z^3\) | \(0\) | \(9\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(z (2x - 5z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(-15x^3 + x^2z - 149z^3\) | \(2.254858\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + z^3\) | \(0.713662\) | \(\infty\) |
\((-1 : -7 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - 7z^3\) | \(0\) | \(9\) |
2-torsion field: 5.1.1132096.2
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1.596240 \) |
Real period: | \( 11.23539 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 9 \) |
Leading coefficient: | \( 0.664236 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(7\) | \(2\) | \(4\) | \(3\) | \(( 1 - T )^{2}\) | |
\(19\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.12.2 | no |
\(3\) | 3.720.4 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{13}}{2}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{13}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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);