Properties

Label 2700.b.324000.1
Conductor $2700$
Discriminant $-324000$
Mordell-Weil group \(\Z/{3}\Z \oplus \Z/{6}\Z\)
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

$y^2 + (x^2 + x)y = x^6 + x^5 + 4x^4 + 2x^3 + 4x^2 + x + 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^6 + x^5z + 4x^4z^2 + 2x^3z^3 + 4x^2z^4 + xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 + 4x^5 + 17x^4 + 10x^3 + 17x^2 + 4x + 4$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); Cmin = HyperellipticCurve(R([1, 1, 4, 2, 4, 1, 1]), R([0, 1, 1])) # minimal equation
 
Copy content magma:R<x> := PolynomialRing(Rationals()); Cmin := HyperellipticCurve(R![1, 1, 4, 2, 4, 1, 1], R![0, 1, 1]); // minimal equation
 
Copy content sage:Csim = HyperellipticCurve(R([4, 4, 17, 10, 17, 4, 4])); Csim # simplified equation
 
Copy content magma:Csim, pi := SimplifiedModel(Cmin); Csim; // simplified equation
 

Invariants

Conductor: \( N \)  \(=\)  \(2700\) \(=\) \( 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Copy content magma:Conductor(LSeries(Cmin)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-324000\) \(=\) \( - 2^{5} \cdot 3^{4} \cdot 5^{3} \)
Copy content magma:Discriminant(Cmin); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(3612\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \cdot 43 \)
\( I_4 \)  \(=\) \(34209\) \(=\)  \( 3^{3} \cdot 7 \cdot 181 \)
\( I_6 \)  \(=\) \(40180527\) \(=\)  \( 3^{2} \cdot 101 \cdot 44203 \)
\( I_{10} \)  \(=\) \(41472000\) \(=\)  \( 2^{12} \cdot 3^{4} \cdot 5^{3} \)
\( J_2 \)  \(=\) \(903\) \(=\)  \( 3 \cdot 7 \cdot 43 \)
\( J_4 \)  \(=\) \(32550\) \(=\)  \( 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 31 \)
\( J_6 \)  \(=\) \(1503900\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 557 \)
\( J_8 \)  \(=\) \(74629800\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 5923 \)
\( J_{10} \)  \(=\) \(324000\) \(=\)  \( 2^{5} \cdot 3^{4} \cdot 5^{3} \)
\( g_1 \)  \(=\) \(7412312704503/4000\)
\( g_2 \)  \(=\) \(5917785517/80\)
\( g_3 \)  \(=\) \(151394271/40\)

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

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

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

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(3\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + xz + 4z^2\) \(=\) \(0,\) \(4y\) \(=\) \(-xz^2 + 8z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(3\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + xz + 4z^2\) \(=\) \(0,\) \(4y\) \(=\) \(-xz^2 + 8z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - xz^2\) \(0\) \(3\)
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) \(2x^2 + xz + 4z^2\) \(=\) \(0,\) \(4y\) \(=\) \(4x^2z + 2xz^2 + 16z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{18 +2 \sqrt{-15}})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 8.245576 \)
Tamagawa product: \( 18 \)
Torsion order:\( 18 \)
Leading coefficient: \( 0.458087 \)
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?
\(2\) \(2\) \(5\) \(2\) \(1^*\) \(( 1 + T )^{2}\) yes
\(3\) \(3\) \(4\) \(3\) \(-1\) \(1 - T\) yes
\(5\) \(2\) \(3\) \(3\) \(-1\) \(( 1 - T )( 1 + T )\) 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 30.a
  Elliptic curve isogeny class 90.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);