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Properties

Label 3584.b.229376.1
Conductor $3584$
Discriminant $-229376$
Mordell-Weil group \(\Z \oplus \Z/{4}\Z\)
Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{CM} \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = -6x^4 + 34x^2 - 68x + 40$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -6x^4z^2 + 34x^2z^4 - 68xz^5 + 40z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 - 21x^4 + 4x^3 + 139x^2 - 270x + 161$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([40, -68, 34, 0, -6]), R([1, 1, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![40, -68, 34, 0, -6], R![1, 1, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([161, -270, 139, 4, -21, 2, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3584\) \(=\) \( 2^{9} \cdot 7 \)
Copy content magma:Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(3584,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-229376\) \(=\) \( - 2^{15} \cdot 7 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(420\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
\( I_4 \)  \(=\) \(3963\) \(=\)  \( 3 \cdot 1321 \)
\( I_6 \)  \(=\) \(638988\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \cdot 7607 \)
\( I_{10} \)  \(=\) \(28\) \(=\)  \( 2^{2} \cdot 7 \)
\( J_2 \)  \(=\) \(1680\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
\( J_4 \)  \(=\) \(75328\) \(=\)  \( 2^{6} \cdot 11 \cdot 107 \)
\( J_6 \)  \(=\) \(-5648384\) \(=\)  \( - 2^{12} \cdot 7 \cdot 197 \)
\( J_8 \)  \(=\) \(-3790898176\) \(=\)  \( - 2^{10} \cdot 13 \cdot 47 \cdot 73 \cdot 83 \)
\( J_{10} \)  \(=\) \(229376\) \(=\)  \( 2^{15} \cdot 7 \)
\( g_1 \)  \(=\) \(58344300000\)
\( g_2 \)  \(=\) \(1557171000\)
\( g_3 \)  \(=\) \(-69501600\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

Copy content sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -4 : 1),\, (2 : -7 : 1),\, (2 : -8 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -4 : 1),\, (2 : -7 : 1),\, (2 : -8 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (2 : -1 : 1),\, (2 : 1 : 1),\, (1 : -4 : 1),\, (1 : 4 : 1)\)

Copy content magma:[C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-8,1],C![2,-7,1]]; // minimal model
 
Copy content magma:[C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,4,1],C![2,-1,1],C![2,1,1]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

Copy content magma:#Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

Copy content 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: \(\Z \oplus \Z/{4}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.059979\) \(\infty\)
\((2 : -7 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-7z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.059979\) \(\infty\)
\((2 : -7 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-7z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((1 : 4 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 + z^3\) \(0.059979\) \(\infty\)
\((2 : 1 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 - 13z^3\) \(0\) \(4\)

2-torsion field: 4.0.392.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.059979 \)
Real period: \( 17.68951 \)
Tamagawa product: \( 8 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.530508 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(9\) \(15\) \(8\) \(-1^*\) \(1\) no
\(7\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 7 T^{2} )\) yes

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?
\(2\) 2.90.1 yes
\(3\) 3.270.2 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 112.a
  Elliptic curve isogeny class 32.a

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(8\) in \(\Z \times \Z [\sqrt{-1}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-1}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);