Properties

Label 2205.i
Number of curves 8
Conductor 2205
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("2205.i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2205.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2205.i1 2205j7 [1, -1, 0, -952569, -357605172] [2] 12288  
2205.i2 2205j5 [1, -1, 0, -59544, -5574717] [2, 2] 6144  
2205.i3 2205j8 [1, -1, 0, -48519, -7711362] [2] 12288  
2205.i4 2205j4 [1, -1, 0, -35289, 2560410] [2] 3072  
2205.i5 2205j3 [1, -1, 0, -4419, -51192] [2, 2] 3072  
2205.i6 2205j2 [1, -1, 0, -2214, 40095] [2, 2] 1536  
2205.i7 2205j1 [1, -1, 0, -9, 1728] [2] 768 \(\Gamma_0(N)\)-optimal
2205.i8 2205j6 [1, -1, 0, 15426, -396495] [2] 6144  

Rank

sage: E.rank()
 

The elliptic curves in class 2205.i have rank \(0\).

Modular form 2205.2.a.i

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.