Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = -x^4 - x^3 + 6x^2 - 5x + 1$ | (homogenize, simplify) |
$y^2 + x^3y = -x^4z^2 - x^3z^3 + 6x^2z^4 - 5xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^4 - 4x^3 + 24x^2 - 20x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -5, 6, -1, -1]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -5, 6, -1, -1], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([4, -20, 24, -4, -4, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(102347\) | \(=\) | \( 7 \cdot 14621 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(716429\) | \(=\) | \( 7^{2} \cdot 14621 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(168\) | \(=\) | \( 2^{3} \cdot 3 \cdot 7 \) |
\( I_4 \) | \(=\) | \(5460\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) |
\( I_6 \) | \(=\) | \(15312\) | \(=\) | \( 2^{4} \cdot 3 \cdot 11 \cdot 29 \) |
\( I_{10} \) | \(=\) | \(2865716\) | \(=\) | \( 2^{2} \cdot 7^{2} \cdot 14621 \) |
\( J_2 \) | \(=\) | \(84\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \) |
\( J_4 \) | \(=\) | \(-616\) | \(=\) | \( - 2^{3} \cdot 7 \cdot 11 \) |
\( J_6 \) | \(=\) | \(20904\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \cdot 67 \) |
\( J_8 \) | \(=\) | \(344120\) | \(=\) | \( 2^{3} \cdot 5 \cdot 7 \cdot 1229 \) |
\( J_{10} \) | \(=\) | \(716429\) | \(=\) | \( 7^{2} \cdot 14621 \) |
\( g_1 \) | \(=\) | \(85349376/14621\) | ||
\( g_2 \) | \(=\) | \(-7451136/14621\) | ||
\( g_3 \) | \(=\) | \(3010176/14621\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -3 : 1)\) | \((-1 : 4 : 1)\) | \((3 : -4 : 2)\) | \((5 : -17 : 2)\) | \((3 : -23 : 2)\) | \((-3 : -25 : 2)\) |
\((7 : -43 : 5)\) | \((-3 : 52 : 2)\) | \((5 : -108 : 2)\) | \((7 : -300 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -3 : 1)\) | \((-1 : 4 : 1)\) | \((3 : -4 : 2)\) | \((5 : -17 : 2)\) | \((3 : -23 : 2)\) | \((-3 : -25 : 2)\) |
\((7 : -43 : 5)\) | \((-3 : 52 : 2)\) | \((5 : -108 : 2)\) | \((7 : -300 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) |
\((-1 : -7 : 1)\) | \((-1 : 7 : 1)\) | \((3 : -19 : 2)\) | \((3 : 19 : 2)\) | \((-3 : -77 : 2)\) | \((-3 : 77 : 2)\) |
\((5 : -91 : 2)\) | \((5 : 91 : 2)\) | \((7 : -257 : 5)\) | \((7 : 257 : 5)\) |
magma: [C![-3,-25,2],C![-3,52,2],C![-1,-3,1],C![-1,4,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1],C![3,-23,2],C![3,-4,2],C![5,-108,2],C![5,-17,2],C![7,-300,5],C![7,-43,5]]; // minimal model
magma: [C![-3,-77,2],C![-3,77,2],C![-1,-7,1],C![-1,7,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,-1,1],C![1,1,0],C![1,1,1],C![3,-19,2],C![3,19,2],C![5,-91,2],C![5,91,2],C![7,-257,5],C![7,257,5]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + z^3\) | \(0.446628\) | \(\infty\) |
\((1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.443492\) | \(\infty\) |
\((-1 : -3 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 2z^3\) | \(0.175500\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + z^3\) | \(0.446628\) | \(\infty\) |
\((1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.443492\) | \(\infty\) |
\((-1 : -3 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 2z^3\) | \(0.175500\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 1 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 2z^3\) | \(0.446628\) | \(\infty\) |
\((1 : 1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.443492\) | \(\infty\) |
\((-1 : -7 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 2xz^2 - 4z^3\) | \(0.175500\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.029282 \) |
Real period: | \( 14.72106 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.862130 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(7\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 5 T + 7 T^{2} )\) | |
\(14621\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 124 T + 14621 T^{2} )\) |
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);