Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + 1)y = x^3 - 3x$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + z^3)y = x^3z^3 - 3xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + x^4 + 6x^3 + 2x^2 - 12x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 0, 1]), R([1, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 0, 1], R![1, 0, 1, 1]);
sage: X = HyperellipticCurve(R([1, -12, 2, 6, 1, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(64237\) | \(=\) | \( 64237 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(64237\) | \(=\) | \( 64237 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(508\) | \(=\) | \( 2^{2} \cdot 127 \) |
\( I_4 \) | \(=\) | \(4489\) | \(=\) | \( 67^{2} \) |
\( I_6 \) | \(=\) | \(4345419\) | \(=\) | \( 3 \cdot 13 \cdot 67 \cdot 1663 \) |
\( I_{10} \) | \(=\) | \(-8222336\) | \(=\) | \( - 2^{7} \cdot 64237 \) |
\( J_2 \) | \(=\) | \(127\) | \(=\) | \( 127 \) |
\( J_4 \) | \(=\) | \(485\) | \(=\) | \( 5 \cdot 97 \) |
\( J_6 \) | \(=\) | \(-49013\) | \(=\) | \( - 23 \cdot 2131 \) |
\( J_8 \) | \(=\) | \(-1614969\) | \(=\) | \( - 3^{2} \cdot 179441 \) |
\( J_{10} \) | \(=\) | \(-64237\) | \(=\) | \( -64237 \) |
\( g_1 \) | \(=\) | \(-33038369407/64237\) | ||
\( g_2 \) | \(=\) | \(-993465755/64237\) | ||
\( g_3 \) | \(=\) | \(790530677/64237\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((-2 : 1 : 1)\) | \((-1 : -2 : 1)\) | \((1 : -2 : 1)\) | \((-2 : 2 : 1)\) | \((-3 : -8 : 2)\) | \((-3 : 9 : 2)\) |
\((-2 : 23 : 3)\) | \((25 : 23 : 3)\) | \((-2 : -54 : 3)\) | \((-23 : 1472 : 12)\) | \((-23 : 2619 : 12)\) | \((25 : -17550 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) |
\((-2 : 1 : 1)\) | \((-1 : -2 : 1)\) | \((1 : -2 : 1)\) | \((-2 : 2 : 1)\) | \((-3 : -8 : 2)\) | \((-3 : 9 : 2)\) |
\((-2 : 23 : 3)\) | \((25 : 23 : 3)\) | \((-2 : -54 : 3)\) | \((-23 : 1472 : 12)\) | \((-23 : 2619 : 12)\) | \((25 : -17550 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) | \((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) | \((-3 : -17 : 2)\) | \((-3 : 17 : 2)\) |
\((-2 : -77 : 3)\) | \((-2 : 77 : 3)\) | \((-23 : -1147 : 12)\) | \((-23 : 1147 : 12)\) | \((25 : -17573 : 3)\) | \((25 : 17573 : 3)\) |
magma: [C![-23,1472,12],C![-23,2619,12],C![-3,-8,2],C![-3,9,2],C![-2,-54,3],C![-2,1,1],C![-2,2,1],C![-2,23,3],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![25,-17550,3],C![25,23,3]]; // minimal model
magma: [C![-23,-1147,12],C![-23,1147,12],C![-3,-17,2],C![-3,17,2],C![-2,-77,3],C![-2,-1,1],C![-2,1,1],C![-2,77,3],C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![25,-17573,3],C![25,17573,3]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.428326\) | \(\infty\) |
\((-2 : 1 : 1) - (1 : -1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.392254\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.271869\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.428326\) | \(\infty\) |
\((-2 : 1 : 1) - (1 : -1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.392254\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.271869\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - 2xz^2 + z^3\) | \(0.428326\) | \(\infty\) |
\((-2 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + 3z^3\) | \(0.392254\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + z^3\) | \(0.271869\) | \(\infty\) |
2-torsion field: 6.2.4111168.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.034994 \) |
Real period: | \( 18.15399 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.635284 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(64237\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 38 T + 64237 T^{2} )\) |
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);