Properties

Label 507.a
Number of curves $4$
Conductor $507$
CM no
Rank $1$
Graph

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

Elliptic curves in class 507.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
507.a1 507c3 \([1, 1, 1, -11749, -495058]\) \(37159393753/1053\) \(5082629877\) \([2]\) \(672\) \(0.96411\)  
507.a2 507c4 \([1, 1, 1, -3299, 64670]\) \(822656953/85683\) \(413575475547\) \([2]\) \(672\) \(0.96411\)  
507.a3 507c2 \([1, 1, 1, -764, -7324]\) \(10218313/1521\) \(7341576489\) \([2, 2]\) \(336\) \(0.61753\)  
507.a4 507c1 \([1, 1, 1, 81, -564]\) \(12167/39\) \(-188245551\) \([4]\) \(168\) \(0.27096\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 507.a have rank \(1\).

Complex multiplication

The elliptic curves in class 507.a do not have complex multiplication.

Modular form 507.2.a.a

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.