Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -5x^4 + 25x^2 - 45$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -5x^4z^2 + 25x^2z^4 - 45z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 18x^4 + 101x^2 - 180$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-45, 0, 25, 0, -5]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-45, 0, 25, 0, -5], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-180, 0, 101, 0, -18, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(600\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(450000\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 5^{5} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(18072\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 251 \) |
\( I_4 \) | \(=\) | \(38904\) | \(=\) | \( 2^{3} \cdot 3 \cdot 1621 \) |
\( I_6 \) | \(=\) | \(233095932\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 6474887 \) |
\( I_{10} \) | \(=\) | \(1800000\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 5^{5} \) |
\( J_2 \) | \(=\) | \(9036\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 251 \) |
\( J_4 \) | \(=\) | \(3395570\) | \(=\) | \( 2 \cdot 5 \cdot 339557 \) |
\( J_6 \) | \(=\) | \(1698206400\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 71 \cdot 151 \) |
\( J_8 \) | \(=\) | \(953774351375\) | \(=\) | \( 5^{3} \cdot 227 \cdot 33613193 \) |
\( J_{10} \) | \(=\) | \(450000\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 5^{5} \) |
\( g_1 \) | \(=\) | \(418329622965299904/3125\) | ||
\( g_2 \) | \(=\) | \(3479436045234936/625\) | ||
\( g_3 \) | \(=\) | \(38515932506304/125\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1),\, (-3 : 15 : 1),\, (3 : -15 : 1)\) |
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1),\, (-3 : 15 : 1),\, (3 : -15 : 1)\) |
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-2 : 0 : 1),\, (2 : 0 : 1),\, (-3 : 0 : 1),\, (3 : 0 : 1)\) |
magma: [C![-3,15,1],C![-2,5,1],C![1,-1,0],C![1,0,0],C![2,-5,1],C![3,-15,1]]; // minimal model
magma: [C![-3,0,1],C![-2,0,1],C![1,-1,0],C![1,1,0],C![2,0,1],C![3,0,1]]; // simplified model
Number of rational Weierstrass points: \(4\)
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: \(\Z/{2}\Z \times \Z/{2}\Z \times \Z/{8}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| ||||||||||||||||||||||||||||||||||||
| ||||||||||||||||||||||||||||||||||||
|
2-torsion field: \(\Q(\sqrt{5}) \)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 8.316290 \) |
Tamagawa product: | \( 32 \) |
Torsion order: | \( 32 \) |
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 |
---|---|---|---|---|---|
\(2\) | \(3\) | \(4\) | \(2\) | \(1 + T + 2 T^{2}\) | |
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(5\) | \(2\) | \(5\) | \(8\) | \(( 1 - T )^{2}\) |
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 15.a5
Elliptic curve 40.a2
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 \(\Q\).