Properties

Label 221760ij
Number of curves $2$
Conductor $221760$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("ij1")
 
E.isogeny_class()
 

Elliptic curves in class 221760ij

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221760.kx1 221760ij1 \([0, 0, 0, -2877132, 1878316144]\) \(13782741913468081/701662500\) \(134089791897600000\) \([2]\) \(4423680\) \(2.3564\) \(\Gamma_0(N)\)-optimal
221760.kx2 221760ij2 \([0, 0, 0, -2721612, 2090383216]\) \(-11666347147400401/3126621093750\) \(-597506595840000000000\) \([2]\) \(8847360\) \(2.7030\)  

Rank

sage: E.rank()
 

The elliptic curves in class 221760ij have rank \(2\).

Complex multiplication

The elliptic curves in class 221760ij do not have complex multiplication.

Modular form 221760.2.a.ij

sage: E.q_eigenform(10)
 
\(q + q^{5} + q^{7} - q^{11} - 4 q^{13} + 4 q^{17} - 8 q^{19} + O(q^{20})\) Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.