Properties

Label 223440bz
Number of curves $4$
Conductor $223440$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bz1")
 
E.isogeny_class()
 

Elliptic curves in class 223440bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
223440.ec4 223440bz1 \([0, 1, 0, 7318624, -16201668876]\) \(89962967236397039/287450726400000\) \(-138519717929916825600000\) \([2]\) \(20736000\) \(3.1236\) \(\Gamma_0(N)\)-optimal
223440.ec3 223440bz2 \([0, 1, 0, -68948896, -189878065420]\) \(75224183150104868881/11219310000000000\) \(5406476706570240000000000\) \([2]\) \(41472000\) \(3.4702\)  
223440.ec2 223440bz3 \([0, 1, 0, -2588348576, -50686206591756]\) \(-3979640234041473454886161/1471455901872240\) \(-709080331875807902760960\) \([2]\) \(103680000\) \(3.9284\)  
223440.ec1 223440bz4 \([0, 1, 0, -41413580896, -3243875203796620]\) \(16300610738133468173382620881/2228489100\) \(1073887289859686400\) \([2]\) \(207360000\) \(4.2749\)  

Rank

sage: E.rank()
 

The elliptic curves in class 223440bz have rank \(0\).

Complex multiplication

The elliptic curves in class 223440bz do not have complex multiplication.

Modular form 223440.2.a.bz

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.