Properties

Label 1815.d
Number of curves 8
Conductor 1815
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1815.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1815.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1815.d1 1815a7 [1, 1, 0, -261362, 51320691] [2] 5120  
1815.d2 1815a5 [1, 1, 0, -16337, 796536] [2, 2] 2560  
1815.d3 1815a8 [1, 1, 0, -13312, 1104481] [2] 5120  
1815.d4 1815a3 [1, 1, 0, -9682, -370751] [2] 1280  
1815.d5 1815a4 [1, 1, 0, -1212, 7011] [2, 2] 1280  
1815.d6 1815a2 [1, 1, 0, -607, -5936] [2, 2] 640  
1815.d7 1815a1 [1, 1, 0, -2, -249] [2] 320 \(\Gamma_0(N)\)-optimal
1815.d8 1815a6 [1, 1, 0, 4233, 58194] [2] 2560  

Rank

sage: E.rank()
 

The elliptic curves in class 1815.d have rank \(1\).

Modular form 1815.2.a.d

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.