Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - 5x^5 + 7x^4 - 5x^2 + 2x$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - 5x^5z + 7x^4z^2 - 5x^2z^4 + 2xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 20x^5 + 28x^4 - 20x^2 + 8x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, -5, 0, 7, -5, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, -5, 0, 7, -5, 1], R![1]);
sage: X = HyperellipticCurve(R([1, 8, -20, 0, 28, -20, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(78193\) | \(=\) | \( 78193 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(78193\) | \(=\) | \( 78193 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(400\) | \(=\) | \( 2^{4} \cdot 5^{2} \) |
\( I_4 \) | \(=\) | \(20836\) | \(=\) | \( 2^{2} \cdot 5209 \) |
\( I_6 \) | \(=\) | \(1612552\) | \(=\) | \( 2^{3} \cdot 17 \cdot 71 \cdot 167 \) |
\( I_{10} \) | \(=\) | \(312772\) | \(=\) | \( 2^{2} \cdot 78193 \) |
\( J_2 \) | \(=\) | \(200\) | \(=\) | \( 2^{3} \cdot 5^{2} \) |
\( J_4 \) | \(=\) | \(-1806\) | \(=\) | \( - 2 \cdot 3 \cdot 7 \cdot 43 \) |
\( J_6 \) | \(=\) | \(32272\) | \(=\) | \( 2^{4} \cdot 2017 \) |
\( J_8 \) | \(=\) | \(798191\) | \(=\) | \( 798191 \) |
\( J_{10} \) | \(=\) | \(78193\) | \(=\) | \( 78193 \) |
\( g_1 \) | \(=\) | \(320000000000/78193\) | ||
\( g_2 \) | \(=\) | \(-14448000000/78193\) | ||
\( g_3 \) | \(=\) | \(1290880000/78193\) |
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);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) |
\((2 : 0 : 1)\) | \((-1 : 2 : 1)\) | \((2 : -1 : 1)\) | \((-1 : -3 : 1)\) | \((3 : 6 : 1)\) | \((3 : -7 : 1)\) |
\((7 : -91 : 9)\) | \((-4 : 104 : 1)\) | \((-4 : -105 : 1)\) | \((7 : -638 : 9)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) |
\((2 : 0 : 1)\) | \((-1 : 2 : 1)\) | \((2 : -1 : 1)\) | \((-1 : -3 : 1)\) | \((3 : 6 : 1)\) | \((3 : -7 : 1)\) |
\((7 : -91 : 9)\) | \((-4 : 104 : 1)\) | \((-4 : -105 : 1)\) | \((7 : -638 : 9)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((2 : -1 : 1)\) | \((2 : 1 : 1)\) | \((-1 : -5 : 1)\) | \((-1 : 5 : 1)\) | \((3 : -13 : 1)\) | \((3 : 13 : 1)\) |
\((-4 : -209 : 1)\) | \((-4 : 209 : 1)\) | \((7 : -547 : 9)\) | \((7 : 547 : 9)\) |
magma: [C![-4,-105,1],C![-4,104,1],C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-1,1],C![2,0,1],C![3,-7,1],C![3,6,1],C![7,-638,9],C![7,-91,9]]; // minimal model
magma: [C![-4,-209,1],C![-4,209,1],C![-1,-5,1],C![-1,5,1],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![2,-1,1],C![2,1,1],C![3,-13,1],C![3,13,1],C![7,-547,9],C![7,547,9]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 9z^3\) | \(0.529190\) | \(\infty\) |
\((1 : -1 : 1) + (2 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.299617\) | \(\infty\) |
\((0 : 0 : 1) + (2 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.255578\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 9z^3\) | \(0.529190\) | \(\infty\) |
\((1 : -1 : 1) + (2 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.299617\) | \(\infty\) |
\((0 : 0 : 1) + (2 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.255578\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -1 : 1) - (1 : -2 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 - 17z^3\) | \(0.529190\) | \(\infty\) |
\((1 : -1 : 1) + (2 : -1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.299617\) | \(\infty\) |
\((0 : 1 : 1) + (2 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.255578\) | \(\infty\) |
2-torsion field: 6.2.5004352.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.037531 \) |
Real period: | \( 21.39178 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.802859 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(78193\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 336 T + 78193 T^{2} )\) |
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);