Properties

Label 165048.o
Number of curves $4$
Conductor $165048$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 165048.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
165048.o1 165048bn3 \([0, -1, 0, -160992, 24307452]\) \(3044193988/85293\) \(12929459282408448\) \([2]\) \(1441792\) \(1.8706\)  
165048.o2 165048bn2 \([0, -1, 0, -23452, -834860]\) \(37642192/13689\) \(518774600837376\) \([2, 2]\) \(720896\) \(1.5240\)  
165048.o3 165048bn1 \([0, -1, 0, -20807, -1148028]\) \(420616192/117\) \(277123184208\) \([2]\) \(360448\) \(1.1774\) \(\Gamma_0(N)\)-optimal
165048.o4 165048bn4 \([0, -1, 0, 71768, -5976740]\) \(269676572/257049\) \(-38965736685118464\) \([2]\) \(1441792\) \(1.8706\)  

Rank

sage: E.rank()
 

The elliptic curves in class 165048.o have rank \(1\).

Complex multiplication

The elliptic curves in class 165048.o do not have complex multiplication.

Modular form 165048.2.a.o

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - 4 q^{7} + q^{9} + q^{13} - 2 q^{15} - 2 q^{17} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.