Properties

Label 495.a
Number of curves $4$
Conductor $495$
CM no
Rank $1$
Graph

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

Elliptic curves in class 495.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
495.a1 495a3 \([1, -1, 1, -533, -4598]\) \(22930509321/6875\) \(5011875\) \([2]\) \(128\) \(0.26364\)  
495.a2 495a4 \([1, -1, 1, -263, 1666]\) \(2749884201/73205\) \(53366445\) \([2]\) \(128\) \(0.26364\)  
495.a3 495a2 \([1, -1, 1, -38, -44]\) \(8120601/3025\) \(2205225\) \([2, 2]\) \(64\) \(-0.082933\)  
495.a4 495a1 \([1, -1, 1, 7, -8]\) \(59319/55\) \(-40095\) \([2]\) \(32\) \(-0.42951\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 495.a have rank \(1\).

Complex multiplication

The elliptic curves in class 495.a do not have complex multiplication.

Modular form 495.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.