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Properties

Label 550a
Number of curves $2$
Conductor $550$
CM no
Rank $1$
Graph

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

Elliptic curves in class 550a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
550.d1 550a1 \([1, 1, 0, -25, 125]\) \(-117649/440\) \(-6875000\) \([]\) \(96\) \(-0.0023865\) \(\Gamma_0(N)\)-optimal
550.d2 550a2 \([1, 1, 0, 225, -3125]\) \(80062991/332750\) \(-5199218750\) \([]\) \(288\) \(0.54692\)  

Rank

sage: E.rank()
 

The elliptic curves in class 550a have rank \(1\).

Complex multiplication

The elliptic curves in class 550a do not have complex multiplication.

Modular form 550.2.a.a

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.