Properties

Label 135240bw
Number of curves $2$
Conductor $135240$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bw1")
 
E.isogeny_class()
 

Elliptic curves in class 135240bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
135240.k2 135240bw1 \([0, -1, 0, -1569976, -789065444]\) \(-3552342505518244/179863605135\) \(-21668631839260277760\) \([2]\) \(4055040\) \(2.4708\) \(\Gamma_0(N)\)-optimal
135240.k1 135240bw2 \([0, -1, 0, -25417296, -49313592180]\) \(7536914291382802562/17961229575\) \(4327671190055270400\) \([2]\) \(8110080\) \(2.8174\)  

Rank

sage: E.rank()
 

The elliptic curves in class 135240bw have rank \(1\).

Complex multiplication

The elliptic curves in class 135240bw do not have complex multiplication.

Modular form 135240.2.a.bw

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + 2 q^{11} + q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.