Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = 8x^3 + 16x^2 + 10x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = 8x^3z^3 + 16x^2z^4 + 10xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^4 + 34x^3 + 65x^2 + 42x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 10, 16, 8]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 10, 16, 8], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([9, 42, 65, 34, 2, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(38052\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 151 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(684936\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 7 \cdot 151 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1348\) | \(=\) | \( 2^{2} \cdot 337 \) |
\( I_4 \) | \(=\) | \(111937\) | \(=\) | \( 7 \cdot 15991 \) |
\( I_6 \) | \(=\) | \(37067937\) | \(=\) | \( 3 \cdot 12355979 \) |
\( I_{10} \) | \(=\) | \(87671808\) | \(=\) | \( 2^{10} \cdot 3^{4} \cdot 7 \cdot 151 \) |
\( J_2 \) | \(=\) | \(337\) | \(=\) | \( 337 \) |
\( J_4 \) | \(=\) | \(68\) | \(=\) | \( 2^{2} \cdot 17 \) |
\( J_6 \) | \(=\) | \(10368\) | \(=\) | \( 2^{7} \cdot 3^{4} \) |
\( J_8 \) | \(=\) | \(872348\) | \(=\) | \( 2^{2} \cdot 218087 \) |
\( J_{10} \) | \(=\) | \(684936\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 7 \cdot 151 \) |
\( g_1 \) | \(=\) | \(4346598285457/684936\) | ||
\( g_2 \) | \(=\) | \(650636801/171234\) | ||
\( g_3 \) | \(=\) | \(1817104/1057\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((-1 : 1 : 1)\) | \((-1 : 0 : 2)\) |
\((0 : -2 : 1)\) | \((-2 : 3 : 1)\) | \((-1 : -3 : 2)\) | \((-3 : 4 : 1)\) | \((-2 : 6 : 1)\) | \((-3 : 25 : 1)\) |
\((-7 : 135 : 12)\) | \((-9 : 343 : 7)\) | \((-9 : 484 : 7)\) | \((-7 : -512 : 12)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((-1 : 1 : 1)\) | \((-1 : 0 : 2)\) |
\((0 : -2 : 1)\) | \((-2 : 3 : 1)\) | \((-1 : -3 : 2)\) | \((-3 : 4 : 1)\) | \((-2 : 6 : 1)\) | \((-3 : 25 : 1)\) |
\((-7 : 135 : 12)\) | \((-9 : 343 : 7)\) | \((-9 : 484 : 7)\) | \((-7 : -512 : 12)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((0 : -3 : 1)\) | \((0 : 3 : 1)\) |
\((-2 : -3 : 1)\) | \((-2 : 3 : 1)\) | \((-1 : -3 : 2)\) | \((-1 : 3 : 2)\) | \((-3 : -21 : 1)\) | \((-3 : 21 : 1)\) |
\((-9 : -141 : 7)\) | \((-9 : 141 : 7)\) | \((-7 : -647 : 12)\) | \((-7 : 647 : 12)\) |
magma: [C![-9,343,7],C![-9,484,7],C![-7,-512,12],C![-7,135,12],C![-3,4,1],C![-3,25,1],C![-2,3,1],C![-2,6,1],C![-1,-3,2],C![-1,0,1],C![-1,0,2],C![-1,1,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-9,-141,7],C![-9,141,7],C![-7,-647,12],C![-7,647,12],C![-3,-21,1],C![-3,21,1],C![-2,-3,1],C![-2,3,1],C![-1,-3,2],C![-1,-1,1],C![-1,3,2],C![-1,1,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 3 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.071858\) | \(\infty\) |
\((-1 : 0 : 2) - (1 : -1 : 0)\) | \(z (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.087692\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 3 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.071858\) | \(\infty\) |
\((-1 : 0 : 2) - (1 : -1 : 0)\) | \(z (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.087692\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : -3 : 1) + (0 : 3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2 + 3z^3\) | \(0.071858\) | \(\infty\) |
\((-1 : 3 : 2) - (1 : -1 : 0)\) | \(z (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.087692\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.004641 \) |
Real period: | \( 17.74398 \) |
Tamagawa product: | \( 12 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.988401 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(3\) | \(3\) | \(1 + T + T^{2}\) | |
\(3\) | \(2\) | \(4\) | \(4\) | \(( 1 + T )^{2}\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 5 T + 7 T^{2} )\) | |
\(151\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 20 T + 151 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.10.1 | no |
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);