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Properties

Label 38.b
Number of curves $2$
Conductor $38$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 38.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38.b1 38b2 \([1, 1, 1, -70, -279]\) \(-37966934881/4952198\) \(-4952198\) \([]\) \(10\) \(0.017785\)  
38.b2 38b1 \([1, 1, 1, 0, 1]\) \(-1/608\) \(-608\) \([5]\) \(2\) \(-0.78693\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 38.b have rank \(0\).

Complex multiplication

The elliptic curves in class 38.b do not have complex multiplication.

Modular form 38.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 4 q^{5} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9} - 4 q^{10} + 2 q^{11} - q^{12} - q^{13} + 3 q^{14} + 4 q^{15} + q^{16} + 3 q^{17} - 2 q^{18} - q^{19} + O(q^{20})\) 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.