Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = x^5 + x^4 - 3x^3 - 5x^2 + 2x + 4$ | (homogenize, simplify) |
$y^2 + x^3y = x^5z + x^4z^2 - 3x^3z^3 - 5x^2z^4 + 2xz^5 + 4z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 4x^4 - 12x^3 - 20x^2 + 8x + 16$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 2, -5, -3, 1, 1]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 2, -5, -3, 1, 1], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([16, 8, -20, -12, 4, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(110848\) | \(=\) | \( 2^{8} \cdot 433 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-221696\) | \(=\) | \( - 2^{9} \cdot 433 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(26\) | \(=\) | \( 2 \cdot 13 \) |
\( I_4 \) | \(=\) | \(1138\) | \(=\) | \( 2 \cdot 569 \) |
\( I_6 \) | \(=\) | \(27176\) | \(=\) | \( 2^{3} \cdot 43 \cdot 79 \) |
\( I_{10} \) | \(=\) | \(866\) | \(=\) | \( 2 \cdot 433 \) |
\( J_2 \) | \(=\) | \(52\) | \(=\) | \( 2^{2} \cdot 13 \) |
\( J_4 \) | \(=\) | \(-2922\) | \(=\) | \( - 2 \cdot 3 \cdot 487 \) |
\( J_6 \) | \(=\) | \(-149092\) | \(=\) | \( - 2^{2} \cdot 37273 \) |
\( J_8 \) | \(=\) | \(-4072717\) | \(=\) | \( - 11 \cdot 370247 \) |
\( J_{10} \) | \(=\) | \(221696\) | \(=\) | \( 2^{9} \cdot 433 \) |
\( g_1 \) | \(=\) | \(742586/433\) | ||
\( g_2 \) | \(=\) | \(-3209817/1732\) | ||
\( g_3 \) | \(=\) | \(-6299137/3464\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((0 : -2 : 1)\) | \((0 : 2 : 1)\) | \((-2 : 2 : 1)\) | \((3 : -5 : 2)\) | \((-2 : 6 : 1)\) | \((-3 : 10 : 2)\) |
\((-3 : 17 : 2)\) | \((3 : -22 : 2)\) | \((1 : 255 : 5)\) | \((1 : -256 : 5)\) | \((-19 : 1534 : 6)\) | \((-19 : 5325 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((0 : -2 : 1)\) | \((0 : 2 : 1)\) | \((-2 : 2 : 1)\) | \((3 : -5 : 2)\) | \((-2 : 6 : 1)\) | \((-3 : 10 : 2)\) |
\((-3 : 17 : 2)\) | \((3 : -22 : 2)\) | \((1 : 255 : 5)\) | \((1 : -256 : 5)\) | \((-19 : 1534 : 6)\) | \((-19 : 5325 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((0 : -4 : 1)\) | \((0 : 4 : 1)\) | \((-2 : -4 : 1)\) | \((-2 : 4 : 1)\) | \((-3 : -7 : 2)\) | \((-3 : 7 : 2)\) |
\((3 : -17 : 2)\) | \((3 : 17 : 2)\) | \((1 : -511 : 5)\) | \((1 : 511 : 5)\) | \((-19 : -3791 : 6)\) | \((-19 : 3791 : 6)\) |
magma: [C![-19,1534,6],C![-19,5325,6],C![-3,10,2],C![-3,17,2],C![-2,2,1],C![-2,6,1],C![-1,0,1],C![-1,1,1],C![0,-2,1],C![0,2,1],C![1,-256,5],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1],C![1,255,5],C![3,-22,2],C![3,-5,2]]; // minimal model
magma: [C![-19,-3791,6],C![-19,3791,6],C![-3,-7,2],C![-3,7,2],C![-2,-4,1],C![-2,4,1],C![-1,-1,1],C![-1,1,1],C![0,-4,1],C![0,4,1],C![1,-511,5],C![1,-1,0],C![1,-1,1],C![1,1,0],C![1,1,1],C![1,511,5],C![3,-17,2],C![3,17,2]]; // 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: \(\Z \oplus \Z \oplus \Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.665908\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.373024\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z\) | \(0.460053\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.665908\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.373024\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z\) | \(0.460053\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2xz^2\) | \(0.665908\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.373024\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 2x^2z\) | \(0.460053\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2xz^2\) | \(0\) | \(2\) |
2-torsion field: 6.2.383977472.4
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 0.101663 \) |
Real period: | \( 18.43470 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.937065 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(9\) | \(2\) | \(1 + 2 T + 2 T^{2}\) | |
\(433\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 433 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.15.1 | yes |
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);