Properties

Label 84525.cn
Number of curves $4$
Conductor $84525$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cn1")
 
E.isogeny_class()
 

Elliptic curves in class 84525.cn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84525.cn1 84525r4 \([1, 1, 0, -6187500, -5926564125]\) \(14251520160844849/264449745\) \(486128875773515625\) \([2]\) \(2211840\) \(2.5187\)  
84525.cn2 84525r2 \([1, 1, 0, -399375, -86346000]\) \(3832302404449/472410225\) \(868415477516015625\) \([2, 2]\) \(1105920\) \(2.1721\)  
84525.cn3 84525r1 \([1, 1, 0, -99250, 10594375]\) \(58818484369/7455105\) \(13704463252265625\) \([2]\) \(552960\) \(1.8256\) \(\Gamma_0(N)\)-optimal
84525.cn4 84525r3 \([1, 1, 0, 586750, -444309375]\) \(12152722588271/53476250625\) \(-98303553277822265625\) \([2]\) \(2211840\) \(2.5187\)  

Rank

sage: E.rank()
 

The elliptic curves in class 84525.cn have rank \(1\).

Complex multiplication

The elliptic curves in class 84525.cn do not have complex multiplication.

Modular form 84525.2.a.cn

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - q^{6} - 3 q^{8} + q^{9} + 4 q^{11} + q^{12} - 6 q^{13} - q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.