Properties

Label 221067.d
Number of curves $2$
Conductor $221067$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 221067.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221067.d1 221067m2 [1, -1, 1, -64080776, 196998840316] [2] 25712640  
221067.d2 221067m1 [1, -1, 1, -2448821, 5496029740] [2] 12856320 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 221067.d have rank \(1\).

Complex multiplication

The elliptic curves in class 221067.d do not have complex multiplication.

Modular form 221067.2.a.d

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} - 2q^{5} + q^{7} + 3q^{8} + 2q^{10} + 2q^{13} - q^{14} - q^{16} + 2q^{17} - 6q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.