Properties

Label 221067z
Number of curves $4$
Conductor $221067$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 221067z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221067.bb3 221067z1 [1, -1, 0, -53928, 4777339] [2] 798720 \(\Gamma_0(N)\)-optimal
221067.bb2 221067z2 [1, -1, 0, -102933, -5209880] [2, 2] 1597440  
221067.bb4 221067z3 [1, -1, 0, 370782, -39980561] [2] 3194880  
221067.bb1 221067z4 [1, -1, 0, -1360728, -610209275] [2] 3194880  

Rank

sage: E.rank()
 

The elliptic curves in class 221067z have rank \(0\).

Complex multiplication

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

Modular form 221067.2.a.z

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.