Properties

Label 1638.f
Number of curves $3$
Conductor $1638$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1638.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.f1 1638j3 \([1, -1, 0, -140967, 20406843]\) \(-424962187484640625/182\) \(-132678\) \([3]\) \(3240\) \(1.2298\)  
1638.f2 1638j2 \([1, -1, 0, -1737, 28485]\) \(-795309684625/6028568\) \(-4394826072\) \([3]\) \(1080\) \(0.68046\)  
1638.f3 1638j1 \([1, -1, 0, 63, 189]\) \(37595375/46592\) \(-33965568\) \([]\) \(360\) \(0.13115\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1638.f have rank \(0\).

Complex multiplication

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

Modular form 1638.2.a.f

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.