Properties

Label 227430.ec
Number of curves $6$
Conductor $227430$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("227430.ec1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 227430.ec

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
227430.ec1 227430be6 [1, -1, 1, -54583268, -155202684343] [2] 14155776  
227430.ec2 227430be4 [1, -1, 1, -3411518, -2424307543] [2, 2] 7077888  
227430.ec3 227430be5 [1, -1, 1, -3184088, -2761631719] [2] 14155776  
227430.ec4 227430be3 [1, -1, 1, -1202198, 479830601] [2] 7077888  
227430.ec5 227430be2 [1, -1, 1, -227498, -32471719] [2, 2] 3538944  
227430.ec6 227430be1 [1, -1, 1, 32422, -3152743] [2] 1769472 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 227430.ec have rank \(1\).

Modular form 227430.2.a.ec

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} - 4q^{11} + 2q^{13} + q^{14} + q^{16} - 2q^{17} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.