Properties

Label 4400.b.352000.1
Conductor $4400$
Discriminant $-352000$
Mordell-Weil group \(\Z \oplus \Z/{3}\Z \oplus \Z/{6}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^6 - 2x^4 - 3x^3 + x^2 + 3x + 1$ (homogenize, simplify)
$y^2 = x^6 - 2x^4z^2 - 3x^3z^3 + x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 - 3x^3 + x^2 + 3x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); Cmin = HyperellipticCurve(R([1, 3, 1, -3, -2, 0, 1]), R([])) # minimal equation
 
Copy content magma:R<x> := PolynomialRing(Rationals()); Cmin := HyperellipticCurve(R![1, 3, 1, -3, -2, 0, 1], R![]); // minimal equation
 
Copy content sage:Csim = HyperellipticCurve(R([1, 3, 1, -3, -2, 0, 1])); Csim # simplified equation
 
Copy content magma:Csim, pi := SimplifiedModel(Cmin); Csim; // simplified equation
 

Invariants

Conductor: \( N \)  \(=\)  \(4400\) \(=\) \( 2^{4} \cdot 5^{2} \cdot 11 \)
Copy content magma:Conductor(LSeries(Cmin)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-352000\) \(=\) \( - 2^{8} \cdot 5^{3} \cdot 11 \)
Copy content magma:Discriminant(Cmin); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(154\) \(=\)  \( 2 \cdot 7 \cdot 11 \)
\( I_4 \)  \(=\) \(1876\) \(=\)  \( 2^{2} \cdot 7 \cdot 67 \)
\( I_6 \)  \(=\) \(128326\) \(=\)  \( 2 \cdot 11 \cdot 19 \cdot 307 \)
\( I_{10} \)  \(=\) \(1375\) \(=\)  \( 5^{3} \cdot 11 \)
\( J_2 \)  \(=\) \(308\) \(=\)  \( 2^{2} \cdot 7 \cdot 11 \)
\( J_4 \)  \(=\) \(-1050\) \(=\)  \( - 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
\( J_6 \)  \(=\) \(-416900\) \(=\)  \( - 2^{2} \cdot 5^{2} \cdot 11 \cdot 379 \)
\( J_8 \)  \(=\) \(-32376925\) \(=\)  \( - 5^{2} \cdot 7 \cdot 17 \cdot 10883 \)
\( J_{10} \)  \(=\) \(352000\) \(=\)  \( 2^{8} \cdot 5^{3} \cdot 11 \)
\( g_1 \)  \(=\) \(984285148/125\)
\( g_2 \)  \(=\) \(-871563/10\)
\( g_3 \)  \(=\) \(-2247091/20\)

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

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

Automorphism group

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

Rational points

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

Copy content magma:[Cmin![-1,-1,1],Cmin![-1,-1,2],Cmin![-1,1,1],Cmin![-1,1,2],Cmin![0,-1,1],Cmin![0,1,1],Cmin![1,-1,0],Cmin![1,-1,1],Cmin![1,1,0],Cmin![1,1,1]]; // minimal model
 
Copy content magma:[Csim![-1,-1,1],Csim![-1,-1,2],Csim![-1,1,1],Csim![-1,1,2],Csim![0,-1,1],Csim![0,1,1],Csim![1,-1,0],Csim![1,-1,1],Csim![1,1,0],Csim![1,1,1]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

Copy content magma:f,h:=HyperellipticPolynomials(Cmin); 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: \(\Z \oplus \Z/{3}\Z \oplus \Z/{6}\Z\)

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

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.365678\) \(\infty\)
\(D_0 - 2 \cdot(1 : 1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0\) \(3\)
\((-1 : -1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.365678\) \(\infty\)
\(D_0 - 2 \cdot(1 : 1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0\) \(3\)
\((-1 : -1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.365678\) \(\infty\)
\(D_0 - 2 \cdot(1 : 1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0\) \(3\)
\((-1 : -1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{-2 -6 \sqrt{5}})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.365678 \)
Real period: \( 17.62720 \)
Tamagawa product: \( 27 \)
Torsion order:\( 18 \)
Leading coefficient: \( 0.537157 \)
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?
\(2\) \(4\) \(8\) \(9\) \(1^*\) \(1\) yes
\(5\) \(2\) \(3\) \(3\) \(-1\) \(( 1 - T )( 1 + T )\) yes
\(11\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 11 T^{2} )\) 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
\(3\) 3.5760.3 yes

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 curve isogeny classes:
  Elliptic curve isogeny class 220.a
  Elliptic curve isogeny class 20.a

Copy content magma:HeuristicDecompositionFactors(Cmin);
 

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 \(\Q\).

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

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

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