Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x + 1)y = 7x^6 + 39x^5 + 2x^4 + 28x^3 - 14x^2 - 8x - 1$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2 + z^3)y = 7x^6 + 39x^5z + 2x^4z^2 + 28x^3z^3 - 14x^2z^4 - 8xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = 28x^6 + 156x^5 + 9x^4 + 114x^3 - 53x^2 - 30x - 3$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -8, -14, 28, 2, 39, 7]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -8, -14, 28, 2, 39, 7], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([-3, -30, -53, 114, 9, 156, 28]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(7140\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-14280\) | \(=\) | \( - 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(40716\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 13 \cdot 29 \) |
| \( I_4 \) | \(=\) | \(-1225313367\) | \(=\) | \( - 3 \cdot 449 \cdot 547 \cdot 1663 \) |
| \( I_6 \) | \(=\) | \(-3692075477589\) | \(=\) | \( - 3^{3} \cdot 1019 \cdot 134193853 \) |
| \( I_{10} \) | \(=\) | \(1827840\) | \(=\) | \( 2^{10} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
| \( J_2 \) | \(=\) | \(10179\) | \(=\) | \( 3^{3} \cdot 13 \cdot 29 \) |
| \( J_4 \) | \(=\) | \(55371892\) | \(=\) | \( 2^{2} \cdot 13842973 \) |
| \( J_6 \) | \(=\) | \(-90637046256\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 199 \cdot 1373 \cdot 6911 \) |
| \( J_8 \) | \(=\) | \(-997160229374872\) | \(=\) | \( - 2^{3} \cdot 277 \cdot 389 \cdot 883 \cdot 1310041 \) |
| \( J_{10} \) | \(=\) | \(14280\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
| \( g_1 \) | \(=\) | \(36425398951350015633/4760\) | ||
| \( g_2 \) | \(=\) | \(4866575441726570949/1190\) | ||
| \( g_3 \) | \(=\) | \(-391295389699815354/595\) |
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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(28x^2 - 12xz - 3z^2\) | \(=\) | \(0,\) | \(56y\) | \(=\) | \(-40xz^2 - 31z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(28x^2 - 12xz - 3z^2\) | \(=\) | \(0,\) | \(56y\) | \(=\) | \(-40xz^2 - 31z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(28x^2 - 12xz - 3z^2\) | \(=\) | \(0,\) | \(56y\) | \(=\) | \(x^2z - 79xz^2 - 61z^3\) | \(0\) | \(2\) |
2-torsion field: 8.4.46982799360000.26
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(4\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.565877 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 1.131754 \) |
| Analytic order of Ш: | \( 8 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(3\) | \(1\) | \(( 1 - T )( 1 + T )\) | |
| \(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) |  |
| \(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
| \(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) |  |
| \(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 17 T^{2} )\) | |
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? |
|---|---|---|
| \(2\) | 2.45.1 | 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);