Properties

Label 338.e
Number of curves $2$
Conductor $338$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 338.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.e1 338b2 \([1, -1, 1, -65773, -6478507]\) \(-38575685889/16384\) \(-13364932132864\) \([]\) \(1092\) \(1.4794\)  
338.e2 338b1 \([1, -1, 1, 137, 2643]\) \(351/4\) \(-3262922884\) \([]\) \(156\) \(0.50645\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 338.e have rank \(0\).

Complex multiplication

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

Modular form 338.2.a.e

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