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Properties

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

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Show commands: SageMath
E = EllipticCurve("eg1")
 
E.isogeny_class()
 

Elliptic curves in class 205350eg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
205350.s2 205350eg1 \([1, 1, 0, -703786500, 7031988594000]\) \(513108539209/12597120\) \(946473662023005218670000000\) \([]\) \(177230592\) \(3.9601\) \(\Gamma_0(N)\)-optimal
205350.s1 205350eg2 \([1, 1, 0, -7029079875, -224100556621875]\) \(511189448451769/7077888000\) \(531790962914434756608000000000\) \([]\) \(531691776\) \(4.5094\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 205350.2.a.eg

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