Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x + 1)y = -x^5 + 3x^4 - 4x^2$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2 + z^3)y = -x^5z + 3x^4z^2 - 4x^2z^4$ | (dehomogenize, simplify) |
| $y^2 = -4x^5 + 13x^4 + 2x^3 - 13x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -4, 0, 3, -1]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -4, 0, 3, -1], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([1, 2, -13, 2, 13, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(107989\) | \(=\) | \( 7 \cdot 15427 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(755923\) | \(=\) | \( 7^{2} \cdot 15427 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(1204\) | \(=\) | \( 2^{2} \cdot 7 \cdot 43 \) |
| \( I_4 \) | \(=\) | \(45817\) | \(=\) | \( 45817 \) |
| \( I_6 \) | \(=\) | \(16295341\) | \(=\) | \( 16295341 \) |
| \( I_{10} \) | \(=\) | \(96758144\) | \(=\) | \( 2^{7} \cdot 7^{2} \cdot 15427 \) |
| \( J_2 \) | \(=\) | \(301\) | \(=\) | \( 7 \cdot 43 \) |
| \( J_4 \) | \(=\) | \(1866\) | \(=\) | \( 2 \cdot 3 \cdot 311 \) |
| \( J_6 \) | \(=\) | \(-3580\) | \(=\) | \( - 2^{2} \cdot 5 \cdot 179 \) |
| \( J_8 \) | \(=\) | \(-1139884\) | \(=\) | \( - 2^{2} \cdot 17 \cdot 16763 \) |
| \( J_{10} \) | \(=\) | \(755923\) | \(=\) | \( 7^{2} \cdot 15427 \) |
| \( g_1 \) | \(=\) | \(50423895949/15427\) | ||
| \( g_2 \) | \(=\) | \(1038520434/15427\) | ||
| \( g_3 \) | \(=\) | \(-6619420/15427\) |
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);
|
Rational points
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
| \((2 : 0 : 1)\) | \((1 : -2 : 1)\) | \((3 : -4 : 1)\) | \((2 : -7 : 1)\) | \((3 : -9 : 1)\) | \((1 : -14 : 4)\) |
| \((-3 : 18 : 1)\) | \((-3 : -25 : 1)\) | \((1 : -70 : 4)\) | \((-10 : 525 : 9)\) | \((-10 : -1344 : 9)\) | |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
| \((2 : 0 : 1)\) | \((1 : -2 : 1)\) | \((3 : -4 : 1)\) | \((2 : -7 : 1)\) | \((3 : -9 : 1)\) | \((1 : -14 : 4)\) |
| \((-3 : 18 : 1)\) | \((-3 : -25 : 1)\) | \((1 : -70 : 4)\) | \((-10 : 525 : 9)\) | \((-10 : -1344 : 9)\) | |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
| \((1 : 1 : 1)\) | \((3 : -5 : 1)\) | \((3 : 5 : 1)\) | \((2 : -7 : 1)\) | \((2 : 7 : 1)\) | \((-3 : -43 : 1)\) |
| \((-3 : 43 : 1)\) | \((1 : -56 : 4)\) | \((1 : 56 : 4)\) | \((-10 : -1869 : 9)\) | \((-10 : 1869 : 9)\) | |
magma: [C![-10,-1344,9],C![-10,525,9],C![-3,-25,1],C![-3,18,1],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-70,4],C![1,-14,4],C![1,-2,1],C![1,-1,1],C![1,0,0],C![2,-7,1],C![2,0,1],C![3,-9,1],C![3,-4,1]]; // minimal model
magma: [C![-10,-1869,9],C![-10,1869,9],C![-3,-43,1],C![-3,43,1],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-56,4],C![1,56,4],C![1,-1,1],C![1,1,1],C![1,0,0],C![2,-7,1],C![2,7,1],C![3,-5,1],C![3,5,1]]; // simplified model
Number of rational Weierstrass points: \(1\)
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 | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(5xz^2 - 3z^3\) | \(0.474918\) | \(\infty\) |
| \((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.271656\) | \(\infty\) |
| \((0 : -1 : 1) + (2 : -7 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - z^3\) | \(0.271802\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(5xz^2 - 3z^3\) | \(0.474918\) | \(\infty\) |
| \((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.271656\) | \(\infty\) |
| \((0 : -1 : 1) + (2 : -7 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - z^3\) | \(0.271802\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 11xz^2 - 5z^3\) | \(0.474918\) | \(\infty\) |
| \((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2 - z^3\) | \(0.271656\) | \(\infty\) |
| \((0 : -1 : 1) + (2 : -7 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - 5xz^2 - z^3\) | \(0.271802\) | \(\infty\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(3\) (upper bound) |
| Mordell-Weil rank: | \(3\) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 0.031585 \) |
| Real period: | \( 15.46537 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.976963 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(7\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 3 T + 7 T^{2} )\) |  |
| \(15427\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 100 T + 15427 T^{2} )\) |  |
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? |
|---|---|---|
| \(2\) | 2.6.1 | 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);