Properties

Label 51984cw
Number of curves $3$
Conductor $51984$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51984cw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51984.i3 51984cw1 \([0, 0, 0, 34656, 109744]\) \(32768/19\) \(-2669086710706176\) \([]\) \(207360\) \(1.6495\) \(\Gamma_0(N)\)-optimal
51984.i2 51984cw2 \([0, 0, 0, -485184, 138387184]\) \(-89915392/6859\) \(-963540302564929536\) \([]\) \(622080\) \(2.1988\)  
51984.i1 51984cw3 \([0, 0, 0, -39993024, 97347427504]\) \(-50357871050752/19\) \(-2669086710706176\) \([]\) \(1866240\) \(2.7481\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51984cw have rank \(0\).

Complex multiplication

The elliptic curves in class 51984cw do not have complex multiplication.

Modular form 51984.2.a.cw

sage: E.q_eigenform(10)
 
\(q - 3q^{5} + q^{7} + 3q^{11} + 4q^{13} + 3q^{17} + O(q^{20})\)  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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.