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Properties

Label 338.b
Number of curves $2$
Conductor $338$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 338.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.b1 338d2 \([1, 1, 0, -54421, 4945517]\) \(-1680914269/32768\) \(-347488235454464\) \([]\) \(1560\) \(1.5836\)  
338.b2 338d1 \([1, 1, 0, 504, -13112]\) \(1331/8\) \(-84835994984\) \([]\) \(312\) \(0.77890\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 338.2.a.b

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.