Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = x^6 - x^4 + x^3 - x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = x^6 - x^4z^2 + x^3z^3 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 3x^4 + 6x^3 + 3x^2 - 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 1, -1, 0, 1]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 1, -1, 0, 1], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([1, -2, 3, 6, -3, 0, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(99361\) | \(=\) | \( 67 \cdot 1483 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-99361\) | \(=\) | \( - 67 \cdot 1483 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(300\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5^{2} \) |
\( I_4 \) | \(=\) | \(26937\) | \(=\) | \( 3^{2} \cdot 41 \cdot 73 \) |
\( I_6 \) | \(=\) | \(1704339\) | \(=\) | \( 3^{2} \cdot 7 \cdot 13 \cdot 2081 \) |
\( I_{10} \) | \(=\) | \(12718208\) | \(=\) | \( 2^{7} \cdot 67 \cdot 1483 \) |
\( J_2 \) | \(=\) | \(75\) | \(=\) | \( 3 \cdot 5^{2} \) |
\( J_4 \) | \(=\) | \(-888\) | \(=\) | \( - 2^{3} \cdot 3 \cdot 37 \) |
\( J_6 \) | \(=\) | \(688\) | \(=\) | \( 2^{4} \cdot 43 \) |
\( J_8 \) | \(=\) | \(-184236\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 13 \cdot 1181 \) |
\( J_{10} \) | \(=\) | \(99361\) | \(=\) | \( 67 \cdot 1483 \) |
\( g_1 \) | \(=\) | \(2373046875/99361\) | ||
\( g_2 \) | \(=\) | \(-374625000/99361\) | ||
\( g_3 \) | \(=\) | \(3870000/99361\) |
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 | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((1 : -3 : 1)\) | \((1 : -7 : 3)\) | \((2 : -7 : 3)\) | \((1 : -32 : 3)\) | \((2 : -50 : 3)\) |
\((-10 : 365 : 7)\) | \((-10 : 378 : 13)\) | \((-10 : -918 : 7)\) | \((-10 : -2185 : 13)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((1 : -3 : 1)\) | \((1 : -7 : 3)\) | \((2 : -7 : 3)\) | \((1 : -32 : 3)\) | \((2 : -50 : 3)\) |
\((-10 : 365 : 7)\) | \((-10 : 378 : 13)\) | \((-10 : -918 : 7)\) | \((-10 : -2185 : 13)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -3 : 1)\) | \((1 : 3 : 1)\) | \((1 : -25 : 3)\) | \((1 : 25 : 3)\) | \((2 : -43 : 3)\) | \((2 : 43 : 3)\) |
\((-10 : -1283 : 7)\) | \((-10 : 1283 : 7)\) | \((-10 : -2563 : 13)\) | \((-10 : 2563 : 13)\) |
magma: [C![-10,-2185,13],C![-10,-918,7],C![-10,365,7],C![-10,378,13],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-32,3],C![1,-7,3],C![1,-3,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-50,3],C![2,-7,3]]; // minimal model
magma: [C![-10,-2563,13],C![-10,-1283,7],C![-10,1283,7],C![-10,2563,13],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-25,3],C![1,25,3],C![1,-3,1],C![1,-2,0],C![1,3,1],C![1,2,0],C![2,-43,3],C![2,43,3]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.354938\) | \(\infty\) |
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.453414\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.381179\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.354938\) | \(\infty\) |
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.453414\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.381179\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2 - z^3\) | \(0.354938\) | \(\infty\) |
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2 - z^3\) | \(0.453414\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 2 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2x^3 + x^2z + xz^2 - z^3\) | \(0.381179\) | \(\infty\) |
2-torsion field: 6.0.6359104.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.058109 \) |
Real period: | \( 16.47041 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.957094 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(67\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 67 T^{2} )\) | |
\(1483\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 44 T + 1483 T^{2} )\) |
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);