Properties

Label 248430.ec
Number of curves $6$
Conductor $248430$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("248430.ec1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 248430.ec

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
248430.ec1 248430ec5 [1, 0, 1, -139120973, 631580989106] [2] 28311552  
248430.ec2 248430ec3 [1, 0, 1, -8695223, 9867524006] [2, 2] 14155776  
248430.ec3 248430ec6 [1, 0, 1, -8115553, 11239950698] [2] 28311552  
248430.ec4 248430ec4 [1, 0, 1, -3064143, -1951516682] [2] 14155776  
248430.ec5 248430ec2 [1, 0, 1, -579843, 132314158] [2, 2] 7077888  
248430.ec6 248430ec1 [1, 0, 1, 82637, 12802766] [2] 3538944 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 248430.ec have rank \(2\).

Modular form 248430.2.a.ec

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.