Properties

Label 18207e
Number of curves 6
Conductor 18207
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("18207.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 18207e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
18207.e6 18207e1 [1, -1, 0, 2547, 10264] [2] 18432 \(\Gamma_0(N)\)-optimal
18207.e5 18207e2 [1, -1, 0, -10458, 90895] [2, 2] 36864  
18207.e3 18207e3 [1, -1, 0, -101493, -12344486] [2] 73728  
18207.e2 18207e4 [1, -1, 0, -127503, 17530600] [2, 2] 73728  
18207.e1 18207e5 [1, -1, 0, -2039238, 1121366389] [2] 147456  
18207.e4 18207e6 [1, -1, 0, -88488, 28431391] [2] 147456  

Rank

sage: E.rank()
 

The elliptic curves in class 18207e have rank \(1\).

Modular form 18207.2.a.e

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} - 2q^{5} + q^{7} - 3q^{8} - 2q^{10} + 4q^{11} - 2q^{13} + q^{14} - q^{16} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.