↑

Properties

Label 196.a.21952.1
Conductor $196$
Discriminant $-21952$
Mordell-Weil group \(\Z/{6}\Z \oplus \Z/{6}\Z\)
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

This is a model for the modular curve $X_0(28)$.

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^6 + 3x^5z + 6x^4z^2 + 7x^3z^3 + 6x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 + 12x^5 + 25x^4 + 30x^3 + 25x^2 + 12x + 4$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 6, 7, 6, 3, 1]), R([0, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 6, 7, 6, 3, 1], R![0, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([4, 12, 25, 30, 25, 12, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(196\) \(=\) \( 2^{2} \cdot 7^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-21952\) \(=\) \( - 2^{6} \cdot 7^{3} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1340\) \(=\)  \( 2^{2} \cdot 5 \cdot 67 \)
\( I_4 \)  \(=\) \(1345\) \(=\)  \( 5 \cdot 269 \)
\( I_6 \)  \(=\) \(149855\) \(=\)  \( 5 \cdot 17 \cdot 41 \cdot 43 \)
\( I_{10} \)  \(=\) \(2809856\) \(=\)  \( 2^{13} \cdot 7^{3} \)
\( J_2 \)  \(=\) \(335\) \(=\)  \( 5 \cdot 67 \)
\( J_4 \)  \(=\) \(4620\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
\( J_6 \)  \(=\) \(90160\) \(=\)  \( 2^{4} \cdot 5 \cdot 7^{2} \cdot 23 \)
\( J_8 \)  \(=\) \(2214800\) \(=\)  \( 2^{4} \cdot 5^{2} \cdot 7^{2} \cdot 113 \)
\( J_{10} \)  \(=\) \(21952\) \(=\)  \( 2^{6} \cdot 7^{3} \)
\( g_1 \)  \(=\) \(4219140959375/21952\)
\( g_2 \)  \(=\) \(6203236875/784\)
\( g_3 \)  \(=\) \(12905875/28\)

  (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

\(\mathrm{Aut}(X)\)\(\simeq\) $D_6$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

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

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

Generator $D_0$ Height Order
\((-1 : 1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0\) \(6\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(3x^2 + 5xz + 4z^2\) \(=\) \(0,\) \(9y\) \(=\) \(8xz^2 + 13z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0\) \(6\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(3x^2 + 5xz + 4z^2\) \(=\) \(0,\) \(9y\) \(=\) \(8xz^2 + 13z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\((-1 : 2 : 1) - (1 : 2 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2x^3 + x^2z + xz^2\) \(0\) \(6\)
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) \(3x^2 + 5xz + 4z^2\) \(=\) \(0,\) \(9y\) \(=\) \(x^2z + 17xz^2 + 26z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{-7}) \)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 11.77714 \)
Tamagawa product: \( 12 \)
Torsion order:\( 36 \)
Leading coefficient: \( 0.109047 \)
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?
\(2\) \(2\) \(6\) \(4\) \(1^*\) \(( 1 + T )^{2}\) yes
\(7\) \(2\) \(3\) \(3\) \(1\) \(( 1 - T )^{2}\) 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?
\(2\) 2.360.3 yes
\(3\) 3.17280.1 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 14.a

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an Eichler order of index \(3\) 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);