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Properties

Label 205350ej
Number of curves $2$
Conductor $205350$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ej1")
 
E.isogeny_class()
 

Elliptic curves in class 205350ej

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
205350.y2 205350ej1 \([1, 1, 0, 2822850, 313807500]\) \(45326591/27000\) \(-1481827269622921875000\) \([]\) \(11508480\) \(2.7516\) \(\Gamma_0(N)\)-optimal
205350.y1 205350ej2 \([1, 1, 0, -35166900, -89076074250]\) \(-87637942369/11718750\) \(-643154196884948730468750\) \([]\) \(34525440\) \(3.3009\)  

Rank

sage: E.rank()
 

The elliptic curves in class 205350ej have rank \(1\).

Complex multiplication

The elliptic curves in class 205350ej do not have complex multiplication.

Modular form 205350.2.a.ej

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