Properties

Label 580.b
Number of curves $2$
Conductor $580$
CM no
Rank $1$
Graph

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

Elliptic curves in class 580.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
580.b1 580b1 \([0, 0, 0, -32, -31]\) \(226492416/105125\) \(1682000\) \([2]\) \(72\) \(-0.11081\) \(\Gamma_0(N)\)-optimal
580.b2 580b2 \([0, 0, 0, 113, -234]\) \(623331504/453125\) \(-116000000\) \([2]\) \(144\) \(0.23576\)  

Rank

sage: E.rank()
 

The elliptic curves in class 580.b have rank \(1\).

Complex multiplication

The elliptic curves in class 580.b do not have complex multiplication.

Modular form 580.2.a.b

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