Properties

Label 2205.c
Number of curves $2$
Conductor $2205$
CM no
Rank $1$
Graph

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

Elliptic curves in class 2205.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.c1 2205d2 \([1, -1, 1, -167, 634]\) \(55306341/15625\) \(144703125\) \([2]\) \(768\) \(0.27272\)  
2205.c2 2205d1 \([1, -1, 1, -62, -164]\) \(2803221/125\) \(1157625\) \([2]\) \(384\) \(-0.073855\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 2205.c have rank \(1\).

Complex multiplication

The elliptic curves in class 2205.c do not have complex multiplication.

Modular form 2205.2.a.c

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.