Properties

Label 18240.cp
Number of curves $4$
Conductor $18240$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 18240.cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
18240.cp1 18240bn3 \([0, 1, 0, -39745, -3058657]\) \(26487576322129/44531250\) \(11673600000000\) \([2]\) \(49152\) \(1.4026\)  
18240.cp2 18240bn2 \([0, 1, 0, -3265, -16225]\) \(14688124849/8122500\) \(2129264640000\) \([2, 2]\) \(24576\) \(1.0561\)  
18240.cp3 18240bn1 \([0, 1, 0, -1985, 33183]\) \(3301293169/22800\) \(5976883200\) \([2]\) \(12288\) \(0.70949\) \(\Gamma_0(N)\)-optimal
18240.cp4 18240bn4 \([0, 1, 0, 12735, -115425]\) \(871257511151/527800050\) \(-138359616307200\) \([4]\) \(49152\) \(1.4026\)  

Rank

sage: E.rank()
 

The elliptic curves in class 18240.cp have rank \(0\).

Complex multiplication

The elliptic curves in class 18240.cp do not have complex multiplication.

Modular form 18240.2.a.cp

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} - 4q^{11} - 2q^{13} + q^{15} + 2q^{17} + q^{19} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.