Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -7x^6 - x^5 - 27x^4 + 27x^3 - 19x^2 - 9x - 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -7x^6 - x^5z - 27x^4z^2 + 27x^3z^3 - 19x^2z^4 - 9xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -27x^6 - 4x^5 - 106x^4 + 110x^3 - 75x^2 - 34x - 3$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -9, -19, 27, -27, -1, -7]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -9, -19, 27, -27, -1, -7], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-3, -34, -75, 110, -106, -4, -27]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(745517\) | \(=\) | \( 745517 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(745517\) | \(=\) | \( 745517 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(34300\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 7^{3} \) |
| \( I_4 \) | \(=\) | \(986710921\) | \(=\) | \( 7 \cdot 7499 \cdot 18797 \) |
| \( I_6 \) | \(=\) | \(7531092661067\) | \(=\) | \( 17 \cdot 233 \cdot 1901310947 \) |
| \( I_{10} \) | \(=\) | \(-95426176\) | \(=\) | \( - 2^{7} \cdot 745517 \) |
| \( J_2 \) | \(=\) | \(8575\) | \(=\) | \( 5^{2} \cdot 7^{3} \) |
| \( J_4 \) | \(=\) | \(-38049179\) | \(=\) | \( - 7 \cdot 17 \cdot 47 \cdot 6803 \) |
| \( J_6 \) | \(=\) | \(-5210185461\) | \(=\) | \( - 3 \cdot 13 \cdot 251 \cdot 532249 \) |
| \( J_8 \) | \(=\) | \(-373104340725529\) | \(=\) | \( - 7^{2} \cdot 151 \cdot 5701 \cdot 8845171 \) |
| \( J_{10} \) | \(=\) | \(-745517\) | \(=\) | \( -745517 \) |
| \( g_1 \) | \(=\) | \(-46362905370537109375/745517\) | ||
| \( g_2 \) | \(=\) | \(23990962750603953125/745517\) | ||
| \( g_3 \) | \(=\) | \(383108193313243125/745517\) |
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
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\Q_{2}$.
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: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
2-torsion field: 6.2.47713088.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.259391 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 9.338093 \) |
| Analytic order of Ш: | \( 36 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(745517\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 1233 T + 745517 T^{2} )\) |  |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .
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);