Properties

Label 18240cb
Number of curves $4$
Conductor $18240$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 18240cb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
18240.bd3 18240cb1 \([0, -1, 0, -1985, -33183]\) \(3301293169/22800\) \(5976883200\) \([2]\) \(12288\) \(0.70949\) \(\Gamma_0(N)\)-optimal
18240.bd2 18240cb2 \([0, -1, 0, -3265, 16225]\) \(14688124849/8122500\) \(2129264640000\) \([2, 2]\) \(24576\) \(1.0561\)  
18240.bd1 18240cb3 \([0, -1, 0, -39745, 3058657]\) \(26487576322129/44531250\) \(11673600000000\) \([4]\) \(49152\) \(1.4026\)  
18240.bd4 18240cb4 \([0, -1, 0, 12735, 115425]\) \(871257511151/527800050\) \(-138359616307200\) \([2]\) \(49152\) \(1.4026\)  

Rank

sage: E.rank()
 

The elliptic curves in class 18240cb have rank \(1\).

Complex multiplication

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

Modular form 18240.2.a.cb

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 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.