Properties

Label 1638.b
Number of curves $2$
Conductor $1638$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1638.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.b1 1638d1 \([1, -1, 0, -21, 63]\) \(-38958219/30758\) \(-830466\) \([3]\) \(288\) \(-0.16566\) \(\Gamma_0(N)\)-optimal
1638.b2 1638d2 \([1, -1, 0, 174, -964]\) \(29503629/35672\) \(-702131976\) \([]\) \(864\) \(0.38364\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1638.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8} + 3 q^{10} - 3 q^{11} + q^{13} - q^{14} + q^{16} + 3 q^{17} + 5 q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.