Properties

Label 34650bc
Number of curves $4$
Conductor $34650$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 34650bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
34650.ba3 34650bc1 \([1, -1, 0, -44644167, 114539415741]\) \(863913648706111516969/2486234429521920\) \(28319764048773120000000\) \([2]\) \(4816896\) \(3.1792\) \(\Gamma_0(N)\)-optimal
34650.ba2 34650bc2 \([1, -1, 0, -63076167, 10970007741]\) \(2436531580079063806249/1405478914998681600\) \(16009283266156857600000000\) \([2, 2]\) \(9633792\) \(3.5258\)  
34650.ba4 34650bc3 \([1, -1, 0, 251851833, 87497511741]\) \(155099895405729262880471/90047655797243760000\) \(-1025699079315479703750000000\) \([2]\) \(19267584\) \(3.8724\)  
34650.ba1 34650bc4 \([1, -1, 0, -672916167, -6694220792259]\) \(2958414657792917260183849/12401051653985258880\) \(141255728996175839430000000\) \([2]\) \(19267584\) \(3.8724\)  

Rank

sage: E.rank()
 

The elliptic curves in class 34650bc have rank \(1\).

Complex multiplication

The elliptic curves in class 34650bc do not have complex multiplication.

Modular form 34650.2.a.bc

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{7} - q^{8} - q^{11} - 6 q^{13} - q^{14} + q^{16} + 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.