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Properties

Label 550.e
Number of curves $3$
Conductor $550$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 550.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
550.e1 550f3 \([1, 0, 1, -758201, 254051548]\) \(-24680042791780949/369098752\) \(-720896000000000\) \([]\) \(6000\) \(1.9878\)  
550.e2 550f1 \([1, 0, 1, -701, -7202]\) \(-19465109/22\) \(-42968750\) \([]\) \(240\) \(0.37841\) \(\Gamma_0(N)\)-optimal
550.e3 550f2 \([1, 0, 1, 4924, 75298]\) \(6761990971/5153632\) \(-10065687500000\) \([]\) \(1200\) \(1.1831\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 550.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9} + q^{11} + q^{12} - 4 q^{13} + 3 q^{14} + q^{16} - 3 q^{17} + 2 q^{18} - 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 25 & 5 \\ 25 & 1 & 5 \\ 5 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.