Properties

Label 441.f
Number of curves 6
Conductor 441
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("441.f1")
sage: E.isogeny_class()

Elliptic curves in class 441.f

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
441.f1 441c5 [1, -1, 0, -345753, -78165914] 2 1536  
441.f2 441c3 [1, -1, 0, -21618, -1216265] 4 768  
441.f3 441c4 [1, -1, 0, -17208, 867901] 2 768  
441.f4 441c6 [1, -1, 0, -15003, -1979636] 2 1536  
441.f5 441c2 [1, -1, 0, -1773, -5720] 4 384  
441.f6 441c1 [1, -1, 0, 432, -869] 2 192 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

The elliptic curves in class 441.f have rank \(1\).

Modular form 441.2.a.f

sage: E.q_eigenform(10)
\( q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.