Properties

Label 13778.a.27556.1
Conductor $13778$
Discriminant $27556$
Mordell-Weil group \(\Z \oplus \Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(13778\) \(=\) \( 2 \cdot 83^{2} \)
Copy content magma:Conductor(LSeries(Cmin)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(27556\) \(=\) \( 2^{2} \cdot 83^{2} \)
Copy content magma:Discriminant(Cmin); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(52\) \(=\)  \( 2^{2} \cdot 13 \)
\( I_4 \)  \(=\) \(5209\) \(=\)  \( 5209 \)
\( I_6 \)  \(=\) \(-16931\) \(=\)  \( -16931 \)
\( I_{10} \)  \(=\) \(3527168\) \(=\)  \( 2^{9} \cdot 83^{2} \)
\( J_2 \)  \(=\) \(13\) \(=\)  \( 13 \)
\( J_4 \)  \(=\) \(-210\) \(=\)  \( - 2 \cdot 3 \cdot 5 \cdot 7 \)
\( J_6 \)  \(=\) \(1024\) \(=\)  \( 2^{10} \)
\( J_8 \)  \(=\) \(-7697\) \(=\)  \( - 43 \cdot 179 \)
\( J_{10} \)  \(=\) \(27556\) \(=\)  \( 2^{2} \cdot 83^{2} \)
\( g_1 \)  \(=\) \(371293/27556\)
\( g_2 \)  \(=\) \(-230685/13778\)
\( g_3 \)  \(=\) \(43264/6889\)

  (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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((-3 : 8 : 1)\) \((-1 : -8 : 3)\) \((-3 : 18 : 1)\) \((-1 : -18 : 3)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((-3 : 8 : 1)\) \((-1 : -8 : 3)\) \((-3 : 18 : 1)\) \((-1 : -18 : 3)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -2 : 1)\) \((1 : 2 : 1)\)
\((-3 : -10 : 1)\) \((-3 : 10 : 1)\) \((-1 : -10 : 3)\) \((-1 : 10 : 3)\)

Copy content magma:[Cmin![-3,8,1],Cmin![-3,18,1],Cmin![-1,-18,3],Cmin![-1,-8,3],Cmin![0,-1,1],Cmin![0,0,1],Cmin![1,-2,1],Cmin![1,-1,0],Cmin![1,0,0],Cmin![1,0,1]]; // minimal model
 
Copy content magma:[Csim![-3,-10,1],Csim![-3,10,1],Csim![-1,-10,3],Csim![-1,10,3],Csim![0,-1,1],Csim![0,1,1],Csim![1,-2,1],Csim![1,-1,0],Csim![1,1,0],Csim![1,2,1]]; // 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: \(\Z \oplus \Z\)

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

Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.133624\) \(\infty\)
\((1 : 0 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.133624\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.133624\) \(\infty\)
\((1 : 0 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.133624\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2xz^2 - z^3\) \(0.133624\) \(\infty\)
\((1 : 2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0.133624\) \(\infty\)

2-torsion field: 3.1.83.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.015948 \)
Real period: \( 16.13052 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.514520 \)
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?
\(2\) \(1\) \(2\) \(2\) \(1^*\) \(( 1 + T )( 1 + T + 2 T^{2} )\) yes
\(83\) \(2\) \(2\) \(1\) \(1\) \(( 1 + T )^{2}\) 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?
\(2\) 2.120.2 no
\(3\) 3.90.1 no

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 83.a
  Elliptic curve isogeny class 166.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);