Properties

Label 86640bz
Number of curves $2$
Conductor $86640$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bz1")
 
E.isogeny_class()
 

Elliptic curves in class 86640bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86640.n2 86640bz1 \([0, -1, 0, -1220661, 519488640]\) \(267219216891904/3655125\) \(2751337212642000\) \([2]\) \(829440\) \(2.1044\) \(\Gamma_0(N)\)-optimal
86640.n1 86640bz2 \([0, -1, 0, -1254956, 488787756]\) \(18148802937424/1947796875\) \(23458769918316000000\) \([2]\) \(1658880\) \(2.4510\)  

Rank

sage: E.rank()
 

The elliptic curves in class 86640bz have rank \(1\).

Complex multiplication

The elliptic curves in class 86640bz do not have complex multiplication.

Modular form 86640.2.a.bz

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