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Properties

Label 84042bu
Number of curves $2$
Conductor $84042$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bu1")
 
E.isogeny_class()
 

Elliptic curves in class 84042bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84042.be2 84042bu1 \([1, -1, 1, -464231, -121629473]\) \(-15177411906818559273/167619938752\) \(-122194935350208\) \([2]\) \(884736\) \(1.8569\) \(\Gamma_0(N)\)-optimal
84042.be1 84042bu2 \([1, -1, 1, -7427711, -7789813649]\) \(62167173500157644301993/7582456\) \(5527610424\) \([2]\) \(1769472\) \(2.2034\)  

Rank

sage: E.rank()
 

The elliptic curves in class 84042bu have rank \(1\).

Complex multiplication

The elliptic curves in class 84042bu do not have complex multiplication.

Modular form 84042.2.a.bu

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 2 q^{5} + q^{7} + q^{8} - 2 q^{10} - 4 q^{11} - 4 q^{13} + q^{14} + q^{16} - 6 q^{19} + O(q^{20})\) 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.