Properties

Label 320b
Number of curves 4
Conductor 320
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("320.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 320b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
320.c3 320b1 [0, 0, 0, -8, 8] [2] 16 \(\Gamma_0(N)\)-optimal
320.c2 320b2 [0, 0, 0, -28, -48] [2, 2] 32  
320.c1 320b3 [0, 0, 0, -428, -3408] [2] 64  
320.c4 320b4 [0, 0, 0, 52, -272] [2] 64  

Rank

sage: E.rank()
 

The elliptic curves in class 320b have rank \(1\).

Modular form 320.2.a.c

sage: E.q_eigenform(10)
 
\( q - q^{5} - 4q^{7} - 3q^{9} - 4q^{11} + 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.