Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1638.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.d1 1638i3 \([1, -1, 0, -3753, -87561]\) \(8020417344913/187278\) \(136525662\) \([2]\) \(1536\) \(0.67328\)  
1638.d2 1638i2 \([1, -1, 0, -243, -1215]\) \(2181825073/298116\) \(217326564\) \([2, 2]\) \(768\) \(0.32670\)  
1638.d3 1638i1 \([1, -1, 0, -63, 189]\) \(38272753/4368\) \(3184272\) \([2]\) \(384\) \(-0.019868\) \(\Gamma_0(N)\)-optimal
1638.d4 1638i4 \([1, -1, 0, 387, -6885]\) \(8780064047/32388174\) \(-23610978846\) \([2]\) \(1536\) \(0.67328\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1638.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} + q^{7} - q^{8} + 2 q^{10} + 4 q^{11} - q^{13} - q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.