Properties

Label 57222k
Number of curves $2$
Conductor $57222$
CM no
Rank $2$
Graph

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

Elliptic curves in class 57222k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
57222.i1 57222k1 [1, -1, 0, -2351358, -1376241260] [2] 2580480 \(\Gamma_0(N)\)-optimal
57222.i2 57222k2 [1, -1, 0, -686718, -3288246764] [2] 5160960  

Rank

sage: E.rank()
 

The elliptic curves in class 57222k have rank \(2\).

Complex multiplication

The elliptic curves in class 57222k do not have complex multiplication.

Modular form 57222.2.a.k

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} - 2q^{5} + 4q^{7} - q^{8} + 2q^{10} - q^{11} - 4q^{13} - 4q^{14} + q^{16} - 8q^{19} + O(q^{20}) \)

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 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.