Properties

Label 272d
Number of curves 4
Conductor 272
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("272.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 272d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
272.d4 272d1 [0, -1, 0, -48, -64] [2] 48 \(\Gamma_0(N)\)-optimal
272.d3 272d2 [0, -1, 0, -688, -6720] [2] 96  
272.d2 272d3 [0, -1, 0, -1648, 26304] [2] 144  
272.d1 272d4 [0, -1, 0, -1808, 21056] [2] 288  

Rank

sage: E.rank()
 

The elliptic curves in class 272d have rank \(0\).

Modular form 272.2.a.d

sage: E.q_eigenform(10)
 
\( q + 2q^{3} + 4q^{7} + q^{9} - 6q^{11} + 2q^{13} - q^{17} + 4q^{19} + O(q^{20}) \)

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.