Properties

Label 40960.d.655360.1
Conductor $40960$
Discriminant $655360$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Sato-Tate group $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - 6x^4 + 6x^3 + 6x^2 + x$ (homogenize, simplify)
$y^2 = x^5z - 6x^4z^2 + 6x^3z^3 + 6x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = x^5 - 6x^4 + 6x^3 + 6x^2 + x$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 6, 6, -6, 1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 6, 6, -6, 1], R![]);
 
sage: X = HyperellipticCurve(R([0, 1, 6, 6, -6, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(40960\) \(=\) \( 2^{13} \cdot 5 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(40960,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(655360\) \(=\) \( 2^{17} \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(416\) \(=\)  \( 2^{5} \cdot 13 \)
\( I_4 \)  \(=\) \(268\) \(=\)  \( 2^{2} \cdot 67 \)
\( I_6 \)  \(=\) \(37184\) \(=\)  \( 2^{6} \cdot 7 \cdot 83 \)
\( I_{10} \)  \(=\) \(80\) \(=\)  \( 2^{4} \cdot 5 \)
\( J_2 \)  \(=\) \(1664\) \(=\)  \( 2^{7} \cdot 13 \)
\( J_4 \)  \(=\) \(112512\) \(=\)  \( 2^{7} \cdot 3 \cdot 293 \)
\( J_6 \)  \(=\) \(9871360\) \(=\)  \( 2^{13} \cdot 5 \cdot 241 \)
\( J_8 \)  \(=\) \(941748224\) \(=\)  \( 2^{12} \cdot 19 \cdot 12101 \)
\( J_{10} \)  \(=\) \(655360\) \(=\)  \( 2^{17} \cdot 5 \)
\( g_1 \)  \(=\) \(97332232192/5\)
\( g_2 \)  \(=\) \(3955021824/5\)
\( g_3 \)  \(=\) \(41706496\)

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);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1)\)

magma: [C![0,0,1],C![1,0,0]]; // minimal model
 
magma: [C![0,0,1],C![1,0,0]]; // 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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 11.80852 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.476065 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(13\) \(17\) \(2\) \(1\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 5 T^{2} )\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.180.3 yes
\(3\) 3.45.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial:
  \(x^{2} + 1\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.0.4.1-2560.1-a
  Elliptic curve isogeny class 2.0.4.1-2560.2-b

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

Of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);