Properties

Label 2205k
Number of curves $4$
Conductor $2205$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2205k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.b3 2205k1 \([1, -1, 1, -1112, 13794]\) \(1771561/105\) \(9005442705\) \([4]\) \(1536\) \(0.66315\) \(\Gamma_0(N)\)-optimal
2205.b2 2205k2 \([1, -1, 1, -3317, -55884]\) \(47045881/11025\) \(945571484025\) \([2, 2]\) \(3072\) \(1.0097\)  
2205.b1 2205k3 \([1, -1, 1, -49622, -4241856]\) \(157551496201/13125\) \(1125680338125\) \([2]\) \(6144\) \(1.3563\)  
2205.b4 2205k4 \([1, -1, 1, 7708, -355764]\) \(590589719/972405\) \(-83399404891005\) \([2]\) \(6144\) \(1.3563\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 2205.2.a.k

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.