Properties

Label 32448cr
Number of curves $2$
Conductor $32448$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cr1")
 
E.isogeny_class()
 

Elliptic curves in class 32448cr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32448.a2 32448cr1 \([0, -1, 0, -3605, -73659]\) \(1048576/117\) \(578290332672\) \([2]\) \(64512\) \(0.98995\) \(\Gamma_0(N)\)-optimal
32448.a1 32448cr2 \([0, -1, 0, -13745, 544881]\) \(3631696/507\) \(40094796398592\) \([2]\) \(129024\) \(1.3365\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32448cr have rank \(0\).

Complex multiplication

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

Modular form 32448.2.a.cr

sage: E.q_eigenform(10)
 
\(q - q^{3} - 4 q^{5} - 2 q^{7} + q^{9} + 4 q^{11} + 4 q^{15} + 2 q^{17} + 2 q^{19} + O(q^{20})\) 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.