Properties

Label 1638.e
Number of curves $4$
Conductor $1638$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1638.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.e1 1638c3 \([1, -1, 0, -1932, -14896]\) \(40530337875/18264064\) \(359491571712\) \([2]\) \(1728\) \(0.91259\)  
1638.e2 1638c1 \([1, -1, 0, -957, 11637]\) \(3592121380875/246064\) \(6643728\) \([6]\) \(576\) \(0.36329\) \(\Gamma_0(N)\)-optimal
1638.e3 1638c2 \([1, -1, 0, -897, 13113]\) \(-2958077788875/946054564\) \(-25543473228\) \([6]\) \(1152\) \(0.70986\)  
1638.e4 1638c4 \([1, -1, 0, 6708, -116848]\) \(1695802078125/1272491584\) \(-25046451847872\) \([2]\) \(3456\) \(1.2592\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1638.e have rank \(1\).

Complex multiplication

The elliptic curves in class 1638.e do not have complex multiplication.

Modular form 1638.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{7} - q^{8} + q^{13} - q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\)  Toggle raw display

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.