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Properties

Label 400.a.409600.1
Conductor $400$
Discriminant $-409600$
Mordell-Weil group \(\Z/{3}\Z \oplus \Z/{6}\Z\)
Sato-Tate group $E_1$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^6 + 4x^4 + 4x^2 + 1$ (homogenize, simplify)
$y^2 = x^6 + 4x^4z^2 + 4x^2z^4 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^4 + 4x^2 + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 0, 4, 0, 4, 0, 1]), R([]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 0, 4, 0, 4, 0, 1], R![]);
 
Copy content sage:X = HyperellipticCurve(R([1, 0, 4, 0, 4, 0, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(400\) \(=\) \( 2^{4} \cdot 5^{2} \)
Copy content magma:Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(400,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-409600\) \(=\) \( - 2^{14} \cdot 5^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(248\) \(=\)  \( 2^{3} \cdot 31 \)
\( I_4 \)  \(=\) \(181\) \(=\)  \( 181 \)
\( I_6 \)  \(=\) \(14873\) \(=\)  \( 107 \cdot 139 \)
\( I_{10} \)  \(=\) \(50\) \(=\)  \( 2 \cdot 5^{2} \)
\( J_2 \)  \(=\) \(992\) \(=\)  \( 2^{5} \cdot 31 \)
\( J_4 \)  \(=\) \(39072\) \(=\)  \( 2^{5} \cdot 3 \cdot 11 \cdot 37 \)
\( J_6 \)  \(=\) \(1945600\) \(=\)  \( 2^{12} \cdot 5^{2} \cdot 19 \)
\( J_8 \)  \(=\) \(100853504\) \(=\)  \( 2^{8} \cdot 151 \cdot 2609 \)
\( J_{10} \)  \(=\) \(409600\) \(=\)  \( 2^{14} \cdot 5^{2} \)
\( g_1 \)  \(=\) \(58632501248/25\)
\( g_2 \)  \(=\) \(2327987904/25\)
\( g_3 \)  \(=\) \(4674304\)

  (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

\(\mathrm{Aut}(X)\)\(\simeq\) $D_4$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_4$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1)\)
All points: \((1 : -1/2 : 0),\, (1 : 1/2 : 0),\, (0 : -1/2 : 1),\, (0 : 1/2 : 1)\)

Copy content magma:[C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]]; // minimal model
 
Copy content magma:[C![0,-1/2,1],C![0,1/2,1],C![1,-1/2,0],C![1,1/2,0]]; // 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: \(\Z/{3}\Z \oplus \Z/{6}\Z\)

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

Generator $D_0$ Height Order
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(2 \cdot(0 : -1/2 : 1) - (1 : -1/2 : 0) - (1 : 1/2 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(-1/2z^3\) \(0\) \(3\)
\((0 : -1/2 : 1) - (1 : 1/2 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-1/2x^3 - 1/2z^3\) \(0\) \(6\)

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

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 7.977094 \)
Tamagawa product: \( 9 \)
Torsion order:\( 18 \)
Leading coefficient: \( 0.221585 \)
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?
\(2\) \(4\) \(14\) \(9\) \(1^*\) \(1\) yes
\(5\) \(2\) \(2\) \(1\) \(1\) \(( 1 + T )^{2}\) 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?
\(2\) 2.180.4 yes
\(3\) 3.17280.4 yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_1$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 20.a

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{}) \otimes \Q\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

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);