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Properties

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

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

Elliptic curves in class 338.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.d1 338e2 \([1, 1, 1, -322, 2127]\) \(-1680914269/32768\) \(-71991296\) \([]\) \(120\) \(0.30114\)  
338.d2 338e1 \([1, 1, 1, 3, -5]\) \(1331/8\) \(-17576\) \([]\) \(24\) \(-0.50358\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 338.2.a.d

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.