Properties

Label 54208cf
Number of curves $2$
Conductor $54208$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cf1")
 
E.isogeny_class()
 

Elliptic curves in class 54208cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54208.cx2 54208cf1 \([0, -1, 0, 26943, 6450017]\) \(4657463/41503\) \(-19274162813796352\) \([2]\) \(368640\) \(1.8061\) \(\Gamma_0(N)\)-optimal
54208.cx1 54208cf2 \([0, -1, 0, -398977, 89674785]\) \(15124197817/1294139\) \(601003440466558976\) \([2]\) \(737280\) \(2.1526\)  

Rank

sage: E.rank()
 

The elliptic curves in class 54208cf have rank \(1\).

Complex multiplication

The elliptic curves in class 54208cf do not have complex multiplication.

Modular form 54208.2.a.cf

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} + 2 q^{5} - q^{7} + q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} + O(q^{20})\) Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.