Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("1470.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1470k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1470.j7 1470k1 [1, 1, 1, 10289, -298411] [4] 6144 \(\Gamma_0(N)\)-optimal
1470.j6 1470k2 [1, 1, 1, -52431, -2731947] [2, 2] 12288  
1470.j4 1470k3 [1, 1, 1, -738431, -244478347] [2] 24576  
1470.j5 1470k4 [1, 1, 1, -369951, 84522549] [2, 2] 24576  
1470.j2 1470k5 [1, 1, 1, -5882451, 5488977549] [2, 2] 49152  
1470.j8 1470k6 [1, 1, 1, 62229, 270705693] [2] 49152  
1470.j1 1470k7 [1, 1, 1, -94119201, 351412332249] [2] 98304  
1470.j3 1470k8 [1, 1, 1, -5845701, 5560992849] [2] 98304  

Rank

sage: E.rank()
 

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

Modular form 1470.2.a.j

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.