Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x)y = 5x^6 - 10x^5 - 7x^4 + 3x^3 + 19x^2 + 20x - 34$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2)y = 5x^6 - 10x^5z - 7x^4z^2 + 3x^3z^3 + 19x^2z^4 + 20xz^5 - 34z^6$ | (dehomogenize, simplify) |
| $y^2 = 20x^6 - 40x^5 - 27x^4 + 14x^3 + 77x^2 + 80x - 136$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-34, 20, 19, 3, -7, -10, 5]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-34, 20, 19, 3, -7, -10, 5], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([-136, 80, 77, 14, -27, -40, 20]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(616510\) | \(=\) | \( 2 \cdot 5 \cdot 61651 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-616510\) | \(=\) | \( - 2 \cdot 5 \cdot 61651 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(279620\) | \(=\) | \( 2^{2} \cdot 5 \cdot 11 \cdot 31 \cdot 41 \) |
| \( I_4 \) | \(=\) | \(-485575103\) | \(=\) | \( - 13 \cdot 23 \cdot 31 \cdot 52387 \) |
| \( I_6 \) | \(=\) | \(-45819761475647\) | \(=\) | \( - 179 \cdot 571 \cdot 448294783 \) |
| \( I_{10} \) | \(=\) | \(-78913280\) | \(=\) | \( - 2^{8} \cdot 5 \cdot 61651 \) |
| \( J_2 \) | \(=\) | \(69905\) | \(=\) | \( 5 \cdot 11 \cdot 31 \cdot 41 \) |
| \( J_4 \) | \(=\) | \(223845172\) | \(=\) | \( 2^{2} \cdot 31 \cdot 1805203 \) |
| \( J_6 \) | \(=\) | \(1034266956876\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 6359 \cdot 1505983 \) |
| \( J_8 \) | \(=\) | \(5548442648176799\) | \(=\) | \( 7 \cdot 31 \cdot 2593 \cdot 9860725079 \) |
| \( J_{10} \) | \(=\) | \(-616510\) | \(=\) | \( - 2 \cdot 5 \cdot 61651 \) |
| \( g_1 \) | \(=\) | \(-333865232753424996188125/123302\) | ||
| \( g_2 \) | \(=\) | \(-7646671826394497865650/61651\) | ||
| \( g_3 \) | \(=\) | \(-505416167242523500590/61651\) |
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 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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(5x^2 - 8z^2\) | \(=\) | \(0,\) | \(10y\) | \(=\) | \(-5xz^2 - 8z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(5x^2 - 8z^2\) | \(=\) | \(0,\) | \(10y\) | \(=\) | \(-5xz^2 - 8z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(5x^2 - 8z^2\) | \(=\) | \(0,\) | \(10y\) | \(=\) | \(x^2z - 9xz^2 - 16z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.243254131264000.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(5\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.807860 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 3.231440 \) |
| Analytic order of Ш: | \( 16 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - T + 2 T^{2} )\) | |
| \(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 3 T + 5 T^{2} )\) |  |
| \(61651\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 268 T + 61651 T^{2} )\) |  |
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? |
|---|---|---|
| \(2\) | 2.15.1 | 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);