Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = 4x^5 + 22x^4 + 46x^3 + 28x^2 + 5x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = 4x^5z + 22x^4z^2 + 46x^3z^3 + 28x^2z^4 + 5xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 16x^5 + 88x^4 + 186x^3 + 112x^2 + 20x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 5, 28, 46, 22, 4]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 5, 28, 46, 22, 4], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, 20, 112, 186, 88, 16, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(5026\) | \(=\) | \( 2 \cdot 7 \cdot 359 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(35182\) | \(=\) | \( 2 \cdot 7^{2} \cdot 359 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(31220\) | \(=\) | \( 2^{2} \cdot 5 \cdot 7 \cdot 223 \) |
\( I_4 \) | \(=\) | \(278329\) | \(=\) | \( 278329 \) |
\( I_6 \) | \(=\) | \(2852760749\) | \(=\) | \( 2852760749 \) |
\( I_{10} \) | \(=\) | \(4503296\) | \(=\) | \( 2^{8} \cdot 7^{2} \cdot 359 \) |
\( J_2 \) | \(=\) | \(7805\) | \(=\) | \( 5 \cdot 7 \cdot 223 \) |
\( J_4 \) | \(=\) | \(2526654\) | \(=\) | \( 2 \cdot 3 \cdot 13 \cdot 29 \cdot 1117 \) |
\( J_6 \) | \(=\) | \(1086135208\) | \(=\) | \( 2^{3} \cdot 113 \cdot 491 \cdot 2447 \) |
\( J_8 \) | \(=\) | \(523326215681\) | \(=\) | \( 41 \cdot 59 \cdot 2153 \cdot 100483 \) |
\( J_{10} \) | \(=\) | \(35182\) | \(=\) | \( 2 \cdot 7^{2} \cdot 359 \) |
\( g_1 \) | \(=\) | \(591110204777028125/718\) | ||
\( g_2 \) | \(=\) | \(12258530733232875/359\) | ||
\( g_3 \) | \(=\) | \(675155221982900/359\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 0 : 2)\) | \((-1 : -7 : 2)\) |
\((-4 : 28 : 1)\) | \((-4 : 35 : 1)\) | \((-6 : 105 : 1)\) | \((-6 : 110 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 0 : 2)\) | \((-1 : -7 : 2)\) |
\((-4 : 28 : 1)\) | \((-4 : 35 : 1)\) | \((-6 : 105 : 1)\) | \((-6 : 110 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-6 : -5 : 1)\) | \((-6 : 5 : 1)\) |
\((-1 : -7 : 2)\) | \((-1 : 7 : 2)\) | \((-4 : -7 : 1)\) | \((-4 : 7 : 1)\) |
magma: [C![-6,105,1],C![-6,110,1],C![-4,28,1],C![-4,35,1],C![-1,-7,2],C![-1,0,2],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-6,-5,1],C![-6,5,1],C![-4,-7,1],C![-4,7,1],C![-1,-7,2],C![-1,7,2],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 5xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-15xz^2 + 3z^3\) | \(0.428002\) | \(\infty\) |
\((-4 : 35 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 4z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-9xz^2 - z^3\) | \(0.075443\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 6xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-35xz^2 - 7z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 5xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-15xz^2 + 3z^3\) | \(0.428002\) | \(\infty\) |
\((-4 : 35 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 4z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-9xz^2 - z^3\) | \(0.075443\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 6xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-35xz^2 - 7z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 5xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 30xz^2 + 7z^3\) | \(0.428002\) | \(\infty\) |
\((-4 : 7 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + 4z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 18xz^2 - z^3\) | \(0.075443\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 6xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 70xz^2 - 13z^3\) | \(0\) | \(2\) |
2-torsion field: 6.6.263948288.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.031070 \) |
Real period: | \( 18.25299 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.283562 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(7\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 3 T + 7 T^{2} )\) | |
\(359\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 12 T + 359 T^{2} )\) |
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);