Properties

Label 25200.ei
Number of curves $8$
Conductor $25200$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("25200.ei1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25200.ei

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25200.ei1 25200ef8 [0, 0, 0, -23223675, 27101214250] [4] 2654208  
25200.ei2 25200ef5 [0, 0, 0, -20739675, 36353898250] [4] 884736  
25200.ei3 25200ef6 [0, 0, 0, -9723675, -11360285750] [2, 2] 1327104  
25200.ei4 25200ef3 [0, 0, 0, -9651675, -11541221750] [2] 663552  
25200.ei5 25200ef2 [0, 0, 0, -1299675, 564858250] [2, 2] 442368  
25200.ei6 25200ef4 [0, 0, 0, -291675, 1418634250] [2] 884736  
25200.ei7 25200ef1 [0, 0, 0, -147675, -7685750] [2] 221184 \(\Gamma_0(N)\)-optimal
25200.ei8 25200ef7 [0, 0, 0, 2624325, -38241881750] [2] 2654208  

Rank

sage: E.rank()
 

The elliptic curves in class 25200.ei have rank \(0\).

Modular form 25200.2.a.ei

sage: E.q_eigenform(10)
 
\( q + q^{7} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.