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Properties

Label 57498.b
Number of curves $2$
Conductor $57498$
CM no
Rank $1$
Graph

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

Elliptic curves in class 57498.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57498.b1 57498c2 \([1, 1, 0, -201271, -34809719]\) \(351447414193/344988\) \(885144822388092\) \([2]\) \(612864\) \(1.7884\)  
57498.b2 57498c1 \([1, 1, 0, -9611, -809235]\) \(-38272753/87024\) \(-223279775016816\) \([2]\) \(306432\) \(1.4418\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 57498.b have rank \(1\).

Complex multiplication

The elliptic curves in class 57498.b do not have complex multiplication.

Modular form 57498.2.a.b

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