Properties

Label 72450.cf
Number of curves $2$
Conductor $72450$
CM no
Rank $1$
Graph

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

Elliptic curves in class 72450.cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.cf1 72450cj1 \([1, -1, 0, -24687, -1338579]\) \(18260010268037/1994194944\) \(181721014272000\) \([2]\) \(294912\) \(1.4698\) \(\Gamma_0(N)\)-optimal
72450.cf2 72450cj2 \([1, -1, 0, 32913, -6695379]\) \(43269428370043/237036554496\) \(-21599956028448000\) \([2]\) \(589824\) \(1.8164\)  

Rank

sage: E.rank()
 

The elliptic curves in class 72450.cf have rank \(1\).

Complex multiplication

The elliptic curves in class 72450.cf do not have complex multiplication.

Modular form 72450.2.a.cf

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{7} - q^{8} + 4 q^{11} - q^{14} + q^{16} + 6 q^{19} + O(q^{20})\) 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 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.