Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -3x^4 + 7x^3 - 4x^2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -3x^4z^2 + 7x^3z^3 - 4x^2z^4$ | (dehomogenize, simplify) |
$y^2 = x^6 - 12x^4 + 30x^3 - 16x^2 + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -4, 7, -3]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -4, 7, -3], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, 0, -16, 30, -12, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(41411\) | \(=\) | \( 41411 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(41411\) | \(=\) | \( 41411 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1044\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 29 \) |
\( I_4 \) | \(=\) | \(16089\) | \(=\) | \( 3 \cdot 31 \cdot 173 \) |
\( I_6 \) | \(=\) | \(2664477\) | \(=\) | \( 3^{2} \cdot 47 \cdot 6299 \) |
\( I_{10} \) | \(=\) | \(5300608\) | \(=\) | \( 2^{7} \cdot 41411 \) |
\( J_2 \) | \(=\) | \(261\) | \(=\) | \( 3^{2} \cdot 29 \) |
\( J_4 \) | \(=\) | \(2168\) | \(=\) | \( 2^{3} \cdot 271 \) |
\( J_6 \) | \(=\) | \(52752\) | \(=\) | \( 2^{4} \cdot 3 \cdot 7 \cdot 157 \) |
\( J_8 \) | \(=\) | \(2267012\) | \(=\) | \( 2^{2} \cdot 11 \cdot 67 \cdot 769 \) |
\( J_{10} \) | \(=\) | \(41411\) | \(=\) | \( 41411 \) |
\( g_1 \) | \(=\) | \(1211162837301/41411\) | ||
\( g_2 \) | \(=\) | \(38546131608/41411\) | ||
\( g_3 \) | \(=\) | \(3593518992/41411\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -2 : 1)\) |
\((2 : -1 : 1)\) | \((1 : -4 : 2)\) | \((4 : 0 : 3)\) | \((1 : -5 : 2)\) | \((2 : -8 : 1)\) | \((2 : -8 : 3)\) |
\((2 : -27 : 3)\) | \((4 : -91 : 3)\) | \((3 : -252 : 10)\) | \((3 : -775 : 10)\) | \((-23 : -525850 : 185)\) | \((-23 : -5793608 : 185)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -2 : 1)\) |
\((2 : -1 : 1)\) | \((1 : -4 : 2)\) | \((4 : 0 : 3)\) | \((1 : -5 : 2)\) | \((2 : -8 : 1)\) | \((2 : -8 : 3)\) |
\((2 : -27 : 3)\) | \((4 : -91 : 3)\) | \((3 : -252 : 10)\) | \((3 : -775 : 10)\) | \((-23 : -525850 : 185)\) | \((-23 : -5793608 : 185)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 2)\) | \((1 : 1 : 2)\) |
\((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((2 : -7 : 1)\) | \((2 : 7 : 1)\) | \((2 : -19 : 3)\) | \((2 : 19 : 3)\) |
\((4 : -91 : 3)\) | \((4 : 91 : 3)\) | \((3 : -523 : 10)\) | \((3 : 523 : 10)\) | \((-23 : -5267758 : 185)\) | \((-23 : 5267758 : 185)\) |
magma: [C![-23,-5793608,185],C![-23,-525850,185],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-27,3],C![2,-8,1],C![2,-8,3],C![2,-1,1],C![3,-775,10],C![3,-252,10],C![4,-91,3],C![4,0,3]]; // minimal model
magma: [C![-23,-5267758,185],C![-23,5267758,185],C![0,-1,1],C![0,1,1],C![1,-1,2],C![1,1,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![2,-19,3],C![2,-7,1],C![2,19,3],C![2,7,1],C![3,-523,10],C![3,523,10],C![4,-91,3],C![4,91,3]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.488627\) | \(\infty\) |
\((1 : -5 : 2) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-11xz^2 + 3z^3\) | \(0.587592\) | \(\infty\) |
\((1 : -5 : 2) - (1 : 0 : 0)\) | \(z (2x - z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-2x^3 - z^3\) | \(0.109268\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.488627\) | \(\infty\) |
\((1 : -5 : 2) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-11xz^2 + 3z^3\) | \(0.587592\) | \(\infty\) |
\((1 : -5 : 2) - (1 : 0 : 0)\) | \(z (2x - z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-2x^3 - z^3\) | \(0.109268\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.488627\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^3 - 22xz^2 + 7z^3\) | \(0.587592\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(z (2x - z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-3x^3 - z^3\) | \(0.109268\) | \(\infty\) |
2-torsion field: 6.2.2650304.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.029860 \) |
Real period: | \( 19.86418 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.593147 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(41411\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 97 T + 41411 T^{2} )\) |
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);