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Properties

Label 42350.bf
Number of curves $2$
Conductor $42350$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 42350.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42350.bf1 42350bl2 \([1, 0, 1, -137701, 27624298]\) \(-417267265/235298\) \(-162829984444531250\) \([]\) \(486000\) \(2.0058\)  
42350.bf2 42350bl1 \([1, 0, 1, 13549, -508202]\) \(397535/392\) \(-271270278125000\) \([]\) \(162000\) \(1.4565\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 42350.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 42350.bf do not have complex multiplication.

Modular form 42350.2.a.bf

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.