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Properties

Label 1638.l
Number of curves $3$
Conductor $1638$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 1638.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.l1 1638t3 \([1, -1, 1, -234509, 43769909]\) \(-1956469094246217097/36641439744\) \(-26711609573376\) \([3]\) \(15552\) \(1.7008\)  
1638.l2 1638t2 \([1, -1, 1, -1094, 133589]\) \(-198461344537/10417365504\) \(-7594259452416\) \([3]\) \(5184\) \(1.1515\)  
1638.l3 1638t1 \([1, -1, 1, 121, -4921]\) \(270840023/14329224\) \(-10446004296\) \([]\) \(1728\) \(0.60215\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1638.2.a.l

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