Properties

Label 578.a
Number of curves $4$
Conductor $578$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 578.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
578.a1 578a4 \([1, 1, 1, -32663, -1583717]\) \(159661140625/48275138\) \(1165244474459522\) \([2]\) \(3456\) \(1.5961\)  
578.a2 578a3 \([1, 1, 1, -29773, -1989473]\) \(120920208625/19652\) \(474351505988\) \([2]\) \(1728\) \(1.2495\)  
578.a3 578a2 \([1, 1, 1, -12433, 528295]\) \(8805624625/2312\) \(55806059528\) \([2]\) \(1152\) \(1.0468\)  
578.a4 578a1 \([1, 1, 1, -873, 5783]\) \(3048625/1088\) \(26261675072\) \([2]\) \(576\) \(0.70021\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 578.a have rank \(0\).

Complex multiplication

The elliptic curves in class 578.a do not have complex multiplication.

Modular form 578.2.a.a

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.