Properties

Label 825.a
Number of curves 4
Conductor 825
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("825.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 825.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
825.a1 825b3 [1, 0, 0, -3663, 84942] [2] 768  
825.a2 825b2 [1, 0, 0, -288, 567] [2, 2] 384  
825.a3 825b1 [1, 0, 0, -163, -808] [2] 192 \(\Gamma_0(N)\)-optimal
825.a4 825b4 [1, 0, 0, 1087, 4692] [2] 768  

Rank

sage: E.rank()
 

The elliptic curves in class 825.a have rank \(1\).

Modular form 825.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} - q^{6} - 4q^{7} + 3q^{8} + q^{9} + q^{11} - q^{12} + 2q^{13} + 4q^{14} - q^{16} + 2q^{17} - q^{18} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.