Properties

Label 32448bz
Number of curves $2$
Conductor $32448$
CM no
Rank $2$
Graph

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

Elliptic curves in class 32448bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32448.t1 32448bz1 \([0, -1, 0, -2253, 19629]\) \(256000/117\) \(578290332672\) \([2]\) \(43008\) \(0.95187\) \(\Gamma_0(N)\)-optimal
32448.t2 32448bz2 \([0, -1, 0, 7887, 139281]\) \(686000/507\) \(-40094796398592\) \([2]\) \(86016\) \(1.2984\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32448bz have rank \(2\).

Complex multiplication

The elliptic curves in class 32448bz do not have complex multiplication.

Modular form 32448.2.a.bz

sage: E.q_eigenform(10)
 
\(q - q^{3} - 4 q^{7} + q^{9} + 2 q^{11} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.