Properties

Label 12615f
Number of curves 8
Conductor 12615
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("12615.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12615f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12615.f7 12615f1 [1, 0, 1, -18, 4543] [2] 6272 \(\Gamma_0(N)\)-optimal
12615.f6 12615f2 [1, 0, 1, -4223, 103781] [2, 2] 12544  
12615.f5 12615f3 [1, 0, 1, -8428, -138427] [2, 2] 25088  
12615.f4 12615f4 [1, 0, 1, -67298, 6714041] [2] 25088  
12615.f2 12615f5 [1, 0, 1, -113553, -14729777] [2, 2] 50176  
12615.f8 12615f6 [1, 0, 1, 29417, -1031569] [2] 50176  
12615.f1 12615f7 [1, 0, 1, -1816578, -942537797] [2] 100352  
12615.f3 12615f8 [1, 0, 1, -92528, -20347657] [2] 100352  

Rank

sage: E.rank()
 

The elliptic curves in class 12615f have rank \(1\).

Modular form 12615.2.a.f

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 3q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{15} - q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.