Properties

Label 720.f
Number of curves $4$
Conductor $720$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 720.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
720.f1 720e3 [0, 0, 0, -1947, -33046] [2] 512  
720.f2 720e4 [0, 0, 0, -1227, 16346] [4] 512  
720.f3 720e2 [0, 0, 0, -147, -286] [2, 2] 256  
720.f4 720e1 [0, 0, 0, 33, -34] [2] 128 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 720.f have rank \(1\).

Complex multiplication

The elliptic curves in class 720.f do not have complex multiplication.

Modular form 720.2.a.f

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.