Properties

Label 2205g
Number of curves $3$
Conductor $2205$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2205g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.e2 2205g1 \([0, 0, 1, -588, 6088]\) \(-262144/35\) \(-3001814235\) \([]\) \(960\) \(0.55111\) \(\Gamma_0(N)\)-optimal
2205.e3 2205g2 \([0, 0, 1, 3822, -15521]\) \(71991296/42875\) \(-3677222437875\) \([]\) \(2880\) \(1.1004\)  
2205.e1 2205g3 \([0, 0, 1, -57918, -5612252]\) \(-250523582464/13671875\) \(-1172583685546875\) \([]\) \(8640\) \(1.6497\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2205g have rank \(1\).

Complex multiplication

The elliptic curves in class 2205g do not have complex multiplication.

Modular form 2205.2.a.g

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.