Properties

Label 338.a
Number of curves $2$
Conductor $338$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 338.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.a1 338f2 \([1, -1, 0, -35944, -2868878]\) \(-1064019559329/125497034\) \(-605750213184506\) \([]\) \(2352\) \(1.5723\)  
338.a2 338f1 \([1, -1, 0, -454, 5812]\) \(-2146689/1664\) \(-8031810176\) \([]\) \(336\) \(0.59931\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 338.a have rank \(1\).

Complex multiplication

The elliptic curves in class 338.a do not have complex multiplication.

Modular form 338.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - 3 q^{3} + q^{4} + q^{5} + 3 q^{6} - q^{7} - q^{8} + 6 q^{9} - q^{10} + 2 q^{11} - 3 q^{12} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} - 6 q^{18} - 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.