Properties

Label 177744.bj
Number of curves $4$
Conductor $177744$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bj1")
 
E.isogeny_class()
 

Elliptic curves in class 177744.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177744.bj1 177744cb4 \([0, -1, 0, -1513778112, -22668969536640]\) \(632678989847546725777/80515134\) \(48820754184782340096\) \([2]\) \(48660480\) \(3.6379\)  
177744.bj2 177744cb3 \([0, -1, 0, -108246272, -245436938112]\) \(231331938231569617/90942310746882\) \(55143324955152100753809408\) \([2]\) \(48660480\) \(3.6379\)  
177744.bj3 177744cb2 \([0, -1, 0, -94619232, -354115307520]\) \(154502321244119857/55101928644\) \(33411330016989612097536\) \([2, 2]\) \(24330240\) \(3.2913\)  
177744.bj4 177744cb1 \([0, -1, 0, -5070112, -7166196992]\) \(-23771111713777/22848457968\) \(-13854276761894967508992\) \([2]\) \(12165120\) \(2.9447\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 177744.bj have rank \(1\).

Complex multiplication

The elliptic curves in class 177744.bj do not have complex multiplication.

Modular form 177744.2.a.bj

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - q^{7} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{17} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.