Minimal equation
Minimal equation
Simplified equation
| $y^2 + x^3y = 3x^4 + 13x^2 + 20$ | (homogenize, simplify) |
| $y^2 + x^3y = 3x^4z^2 + 13x^2z^4 + 20z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 12x^4 + 52x^2 + 80$ | (minimize, homogenize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([20, 0, 13, 0, 3]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![20, 0, 13, 0, 3], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([80, 0, 52, 0, 12, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(640\) | \(=\) | \( 2^{7} \cdot 5 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(640,2),R![1]>*])); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-81920\) | \(=\) | \( - 2^{14} \cdot 5 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(-29184\) | \(=\) | \( - 2^{9} \cdot 3 \cdot 19 \) |
| \( I_4 \) | \(=\) | \(150528\) | \(=\) | \( 2^{10} \cdot 3 \cdot 7^{2} \) |
| \( I_6 \) | \(=\) | \(-1460207616\) | \(=\) | \( - 2^{16} \cdot 3 \cdot 7 \cdot 1061 \) |
| \( I_{10} \) | \(=\) | \(-335544320\) | \(=\) | \( - 2^{26} \cdot 5 \) |
| \( J_2 \) | \(=\) | \(-3648\) | \(=\) | \( - 2^{6} \cdot 3 \cdot 19 \) |
| \( J_4 \) | \(=\) | \(552928\) | \(=\) | \( 2^{5} \cdot 37 \cdot 467 \) |
| \( J_6 \) | \(=\) | \(-111431680\) | \(=\) | \( - 2^{12} \cdot 5 \cdot 5441 \) |
| \( J_8 \) | \(=\) | \(25193348864\) | \(=\) | \( 2^{8} \cdot 73 \cdot 379 \cdot 3557 \) |
| \( J_{10} \) | \(=\) | \(-81920\) | \(=\) | \( - 2^{14} \cdot 5 \) |
| \( g_1 \) | \(=\) | \(39432490647552/5\) | ||
| \( g_2 \) | \(=\) | \(1638374321664/5\) | ||
| \( g_3 \) | \(=\) | \(18102076416\) |
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: \((1 : 0 : 0),\, (1 : -1 : 0)\)
magma: [C![1,-1,0],C![1,0,0]];
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: \(\Z/{12}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + xz + 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - 2z^3\) | \(0\) | \(12\) |
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 7.405674 \) |
| Tamagawa product: | \( 6 \) |
| Torsion order: | \( 12 \) |
| Leading coefficient: | \( 0.308569 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(7\) | \(14\) | \(6\) | \(1\) | |
| \(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 5 T^{2} )\) |  |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $N(G_{1,3})$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 32.a4
Elliptic curve 20.a4
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(4\) in \(\Z \times \Z [\sqrt{-1}]\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-1}) \) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |