Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -2x^4 + x^3 + 2x^2 - 3x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -2x^4z^2 + x^3z^3 + 2x^2z^4 - 3xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 8x^4 + 6x^3 + 8x^2 - 12x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -3, 2, 1, -2]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -3, 2, 1, -2], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([9, -12, 8, 6, -8, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(93162\) | \(=\) | \( 2 \cdot 3 \cdot 15527 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-558972\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 15527 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(460\) | \(=\) | \( 2^{2} \cdot 5 \cdot 23 \) |
\( I_4 \) | \(=\) | \(39913\) | \(=\) | \( 167 \cdot 239 \) |
\( I_6 \) | \(=\) | \(11017259\) | \(=\) | \( 11 \cdot 1001569 \) |
\( I_{10} \) | \(=\) | \(71548416\) | \(=\) | \( 2^{9} \cdot 3^{2} \cdot 15527 \) |
\( J_2 \) | \(=\) | \(115\) | \(=\) | \( 5 \cdot 23 \) |
\( J_4 \) | \(=\) | \(-1112\) | \(=\) | \( - 2^{3} \cdot 139 \) |
\( J_6 \) | \(=\) | \(-96372\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 2677 \) |
\( J_8 \) | \(=\) | \(-3079831\) | \(=\) | \( -3079831 \) |
\( J_{10} \) | \(=\) | \(558972\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 15527 \) |
\( g_1 \) | \(=\) | \(20113571875/558972\) | ||
\( g_2 \) | \(=\) | \(-422803250/139743\) | ||
\( g_3 \) | \(=\) | \(-35403325/15527\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((0 : -2 : 1)\) | \((-1 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((2 : -4 : 1)\) | \((2 : -5 : 1)\) | \((-3 : 10 : 1)\) | \((-4 : 10 : 1)\) |
\((-3 : 16 : 1)\) | \((3 : -16 : 2)\) | \((3 : -19 : 2)\) | \((5 : -37 : 2)\) | \((-4 : 53 : 1)\) | \((5 : -96 : 2)\) |
\((15 : 12256 : 28)\) | \((15 : -37583 : 28)\) | \((-43 : 179982 : 45)\) | \((-43 : -191600 : 45)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((0 : -2 : 1)\) | \((-1 : -2 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((2 : -4 : 1)\) | \((2 : -5 : 1)\) | \((-3 : 10 : 1)\) | \((-4 : 10 : 1)\) |
\((-3 : 16 : 1)\) | \((3 : -16 : 2)\) | \((3 : -19 : 2)\) | \((5 : -37 : 2)\) | \((-4 : 53 : 1)\) | \((5 : -96 : 2)\) |
\((15 : 12256 : 28)\) | \((15 : -37583 : 28)\) | \((-43 : 179982 : 45)\) | \((-43 : -191600 : 45)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((2 : -1 : 1)\) | \((2 : 1 : 1)\) |
\((0 : -3 : 1)\) | \((0 : 3 : 1)\) | \((3 : -3 : 2)\) | \((3 : 3 : 2)\) | \((-1 : -4 : 1)\) | \((-1 : 4 : 1)\) |
\((-3 : -6 : 1)\) | \((-3 : 6 : 1)\) | \((-4 : -43 : 1)\) | \((-4 : 43 : 1)\) | \((5 : -59 : 2)\) | \((5 : 59 : 2)\) |
\((15 : -49839 : 28)\) | \((15 : 49839 : 28)\) | \((-43 : -371582 : 45)\) | \((-43 : 371582 : 45)\) |
magma: [C![-43,-191600,45],C![-43,179982,45],C![-4,10,1],C![-4,53,1],C![-3,10,1],C![-3,16,1],C![-1,-2,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,-4,1],C![3,-19,2],C![3,-16,2],C![5,-96,2],C![5,-37,2],C![15,-37583,28],C![15,12256,28]]; // minimal model
magma: [C![-43,-371582,45],C![-43,371582,45],C![-4,-43,1],C![-4,43,1],C![-3,-6,1],C![-3,6,1],C![-1,-4,1],C![-1,4,1],C![0,-3,1],C![0,3,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![2,-1,1],C![2,1,1],C![3,-3,2],C![3,3,2],C![5,-59,2],C![5,59,2],C![15,-49839,28],C![15,49839,28]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.396197\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.202076\) | \(\infty\) |
\((0 : 1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.202984\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.396197\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.202076\) | \(\infty\) |
\((0 : 1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.202984\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0.396197\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3z^3\) | \(0.202076\) | \(\infty\) |
\((0 : 3 : 1) + (1 : 2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2xz^2 + 3z^3\) | \(0.202984\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.014247 \) |
Real period: | \( 16.00424 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.912113 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(15527\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 99 T + 15527 T^{2} )\) |
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);