Properties

Label 2240k
Number of curves $4$
Conductor $2240$
CM no
Rank $1$
Graph

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

Elliptic curves in class 2240k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2240.p3 2240k1 \([0, 0, 0, -47, 124]\) \(179406144/35\) \(2240\) \([2]\) \(192\) \(-0.35741\) \(\Gamma_0(N)\)-optimal
2240.p2 2240k2 \([0, 0, 0, -52, 96]\) \(3796416/1225\) \(5017600\) \([2, 2]\) \(384\) \(-0.010833\)  
2240.p1 2240k3 \([0, 0, 0, -332, -2256]\) \(123505992/4375\) \(143360000\) \([2]\) \(768\) \(0.33574\)  
2240.p4 2240k4 \([0, 0, 0, 148, 656]\) \(10941048/12005\) \(-393379840\) \([4]\) \(768\) \(0.33574\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2240k have rank \(1\).

Complex multiplication

The elliptic curves in class 2240k do not have complex multiplication.

Modular form 2240.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{5} + q^{7} - 3 q^{9} - 4 q^{11} - 2 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.