Properties

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

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

Elliptic curves in class 2205f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.d2 2205f1 \([0, 0, 1, 42, -212]\) \(229376/675\) \(-24111675\) \([]\) \(384\) \(0.099916\) \(\Gamma_0(N)\)-optimal
2205.d1 2205f2 \([0, 0, 1, -1848, -30641]\) \(-19539165184/46875\) \(-1674421875\) \([]\) \(1152\) \(0.64922\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 2205.2.a.f

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.