Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("21780.q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21780y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21780.q3 21780y1 [0, 0, 0, -1452, 14641] [2] 17280 \(\Gamma_0(N)\)-optimal
21780.q4 21780y2 [0, 0, 0, 3993, 98494] [2] 34560  
21780.q1 21780y3 [0, 0, 0, -45012, -3674891] [2] 51840  
21780.q2 21780y4 [0, 0, 0, -39567, -4597274] [2] 103680  

Rank

sage: E.rank()
 

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

Modular form 21780.2.a.q

sage: E.q_eigenform(10)
 
\( q + q^{5} - 2q^{7} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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.