Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = -x^4 + x^3 - x^2 - 3x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + x^3z^3 - x^2z^4 - 3xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 2x^4 + 6x^3 - 3x^2 - 10x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -3, -1, 1, -1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -3, -1, 1, -1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([9, -10, -3, 6, -2, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(79337\) | \(=\) | \( 79337 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(79337\) | \(=\) | \( 79337 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1020\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 17 \) |
\( I_4 \) | \(=\) | \(28713\) | \(=\) | \( 3 \cdot 17 \cdot 563 \) |
\( I_6 \) | \(=\) | \(10231131\) | \(=\) | \( 3 \cdot 59 \cdot 57803 \) |
\( I_{10} \) | \(=\) | \(-10155136\) | \(=\) | \( - 2^{7} \cdot 79337 \) |
\( J_2 \) | \(=\) | \(255\) | \(=\) | \( 3 \cdot 5 \cdot 17 \) |
\( J_4 \) | \(=\) | \(1513\) | \(=\) | \( 17 \cdot 89 \) |
\( J_6 \) | \(=\) | \(-18973\) | \(=\) | \( -18973 \) |
\( J_8 \) | \(=\) | \(-1781821\) | \(=\) | \( - 17 \cdot 281 \cdot 373 \) |
\( J_{10} \) | \(=\) | \(-79337\) | \(=\) | \( -79337 \) |
\( g_1 \) | \(=\) | \(-1078203909375/79337\) | ||
\( g_2 \) | \(=\) | \(-25087620375/79337\) | ||
\( g_3 \) | \(=\) | \(1233719325/79337\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) | \((0 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((-2 : 5 : 1)\) | \((7 : -23 : 2)\) | \((4 : -37 : 3)\) |
\((4 : -90 : 3)\) | \((7 : -356 : 2)\) | \((11 : -1206 : 15)\) | \((11 : -5975 : 15)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) | \((0 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((-2 : 5 : 1)\) | \((7 : -23 : 2)\) | \((4 : -37 : 3)\) |
\((4 : -90 : 3)\) | \((7 : -356 : 2)\) | \((11 : -1206 : 15)\) | \((11 : -5975 : 15)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) |
\((0 : -3 : 1)\) | \((0 : 3 : 1)\) | \((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) | \((4 : -53 : 3)\) | \((4 : 53 : 3)\) |
\((7 : -333 : 2)\) | \((7 : 333 : 2)\) | \((11 : -4769 : 15)\) | \((11 : 4769 : 15)\) |
magma: [C![-2,4,1],C![-2,5,1],C![-1,-1,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![4,-90,3],C![4,-37,3],C![7,-356,2],C![7,-23,2],C![11,-5975,15],C![11,-1206,15]]; // minimal model
magma: [C![-2,-1,1],C![-2,1,1],C![-1,-3,1],C![-1,3,1],C![0,-3,1],C![0,3,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![4,-53,3],C![4,53,3],C![7,-333,2],C![7,333,2],C![11,-4769,15],C![11,4769,15]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 4 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0.560794\) | \(\infty\) |
\((1 : -2 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.349279\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.236667\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 4 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0.560794\) | \(\infty\) |
\((1 : -2 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.349279\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.236667\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3xz^2 + z^3\) | \(0.560794\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 - z^3\) | \(0.349279\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.236667\) | \(\infty\) |
2-torsion field: 6.2.5077568.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.039916 \) |
Real period: | \( 18.56975 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.741242 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(79337\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 495 T + 79337 T^{2} )\) |
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);