Properties

Label 1470.d
Number of curves 8
Conductor 1470
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1470.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1470.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1470.d1 1470d8 [1, 1, 0, -261342, 51314796] [2] 6912  
1470.d2 1470d7 [1, 1, 0, -22222, 164284] [2] 6912  
1470.d3 1470d6 [1, 1, 0, -16342, 795796] [2, 2] 3456  
1470.d4 1470d4 [1, 1, 0, -14137, -652889] [2] 2304  
1470.d5 1470d5 [1, 1, 0, -3357, 63099] [2] 2304  
1470.d6 1470d2 [1, 1, 0, -907, -9911] [2, 2] 1152  
1470.d7 1470d3 [1, 1, 0, -662, 21204] [2] 1728  
1470.d8 1470d1 [1, 1, 0, 73, -699] [2] 576 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1470.d have rank \(1\).

Modular form 1470.2.a.d

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.