Properties

Label 54150.by
Number of curves $4$
Conductor $54150$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("by1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 54150.by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54150.by1 54150bu4 \([1, 1, 1, -5604713, 5097385781]\) \(26487576322129/44531250\) \(32734560754394531250\) \([2]\) \(2211840\) \(2.6399\)  
54150.by2 54150bu2 \([1, 1, 1, -460463, 25155281]\) \(14688124849/8122500\) \(5970783881601562500\) \([2, 2]\) \(1105920\) \(2.2933\)  
54150.by3 54150bu1 \([1, 1, 1, -279963, -56791719]\) \(3301293169/22800\) \(16760095106250000\) \([2]\) \(552960\) \(1.9467\) \(\Gamma_0(N)\)-optimal
54150.by4 54150bu3 \([1, 1, 1, 1795787, 201142781]\) \(871257511151/527800050\) \(-387981536626469531250\) \([2]\) \(2211840\) \(2.6399\)  

Rank

sage: E.rank()
 

The elliptic curves in class 54150.by have rank \(1\).

Complex multiplication

The elliptic curves in class 54150.by do not have complex multiplication.

Modular form 54150.2.a.by

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