Properties

Label 21780x
Number of curves $4$
Conductor $21780$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("x1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21780x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21780.r4 21780x1 \([0, 0, 0, 238128, -164337239]\) \(72268906496/606436875\) \(-12531100788527310000\) \([2]\) \(276480\) \(2.3459\) \(\Gamma_0(N)\)-optimal
21780.r3 21780x2 \([0, 0, 0, -3437247, -2258565914]\) \(13584145739344/1195803675\) \(395351588729596435200\) \([2]\) \(552960\) \(2.6925\)  
21780.r2 21780x3 \([0, 0, 0, -17011632, -27027819731]\) \(-26348629355659264/24169921875\) \(-499434878636718750000\) \([2]\) \(829440\) \(2.8952\)  
21780.r1 21780x4 \([0, 0, 0, -272246007, -1728981679106]\) \(6749703004355978704/5671875\) \(1875211490988000000\) \([2]\) \(1658880\) \(3.2418\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21780x have rank \(1\).

Complex multiplication

The elliptic curves in class 21780x do not have complex multiplication.

Modular form 21780.2.a.x

sage: E.q_eigenform(10)
 
\(q + q^{5} - 2q^{7} - 2q^{13} - 2q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.