Properties

Label 1296.k
Number of curves $2$
Conductor $1296$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1296.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1296.k1 1296f2 \([0, 0, 0, -81, 243]\) \(6912\) \(8503056\) \([]\) \(216\) \(0.054953\)  
1296.k2 1296f1 \([0, 0, 0, -21, -37]\) \(790272\) \(1296\) \([]\) \(72\) \(-0.49435\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1296.k have rank \(0\).

Complex multiplication

The elliptic curves in class 1296.k do not have complex multiplication.

Modular form 1296.2.a.k

sage: E.q_eigenform(10)
 
\(q + 3q^{5} + q^{7} - 3q^{11} - q^{13} + 6q^{17} + 4q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.