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Properties

Label 479370.bf
Number of curves $2$
Conductor $479370$
CM no
Rank $1$
Graph

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

Elliptic curves in class 479370.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
479370.bf1 479370bf2 \([1, 0, 1, -25248, -1521872]\) \(2992209121/54150\) \(32209682832150\) \([2]\) \(2408448\) \(1.3872\) \(\Gamma_0(N)\)-optimal*
479370.bf2 479370bf1 \([1, 0, 1, -18, -68624]\) \(-1/3420\) \(-2034295757820\) \([2]\) \(1204224\) \(1.0406\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 479370.bf1.

Rank

sage: E.rank()
 

The elliptic curves in class 479370.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 479370.bf do not have complex multiplication.

Modular form 479370.2.a.bf

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.