Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = x^4 + x^2 - 3$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = x^4z^2 + x^2z^4 - 3z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 6x^4 + 5x^2 - 12$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 0, 1, 0, 1]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 0, 1, 0, 1], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-12, 0, 5, 0, 6, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(600\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(30000\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5^{4} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(600\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5^{2} \) |
\( I_4 \) | \(=\) | \(18744\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \cdot 71 \) |
\( I_6 \) | \(=\) | \(4690524\) | \(=\) | \( 2^{2} \cdot 3 \cdot 390877 \) |
\( I_{10} \) | \(=\) | \(120000\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5^{4} \) |
\( J_2 \) | \(=\) | \(300\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5^{2} \) |
\( J_4 \) | \(=\) | \(626\) | \(=\) | \( 2 \cdot 313 \) |
\( J_6 \) | \(=\) | \(-198336\) | \(=\) | \( - 2^{6} \cdot 3 \cdot 1033 \) |
\( J_8 \) | \(=\) | \(-14973169\) | \(=\) | \( - 131 \cdot 114299 \) |
\( J_{10} \) | \(=\) | \(30000\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5^{4} \) |
\( g_1 \) | \(=\) | \(81000000\) | ||
\( g_2 \) | \(=\) | \(563400\) | ||
\( g_3 \) | \(=\) | \(-595008\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 1 : 1),\, (1 : -1 : 1)\) |
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 1 : 1),\, (1 : -1 : 1)\) |
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1)\) |
magma: [C![-1,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
magma: [C![-1,0,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // simplified model
Number of rational Weierstrass points: \(2\)
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: \(\Z/{2}\Z \times \Z/{8}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
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2-torsion field: \(\Q(\zeta_{12})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 8.316290 \) |
Tamagawa product: | \( 8 \) |
Torsion order: | \( 16 \) |
Leading coefficient: | \( 0.259884 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
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\(2\) | \(3\) | \(4\) | \(2\) | \(1 + T + 2 T^{2}\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(5\) | \(2\) | \(4\) | \(4\) | \(( 1 - T )^{2}\) |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 40.a2
Elliptic curve 15.a6
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).