Properties

Label 17689.f
Number of curves $3$
Conductor $17689$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 17689.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
17689.f1 17689g3 \([0, 1, 1, -13608737, -19327549923]\) \(-50357871050752/19\) \(-105163116221611\) \([]\) \(408240\) \(2.4786\)  
17689.f2 17689g2 \([0, 1, 1, -165097, -27524248]\) \(-89915392/6859\) \(-37963884956001571\) \([]\) \(136080\) \(1.9293\)  
17689.f3 17689g1 \([0, 1, 1, 11793, -17853]\) \(32768/19\) \(-105163116221611\) \([]\) \(45360\) \(1.3800\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 17689.f have rank \(1\).

Complex multiplication

The elliptic curves in class 17689.f do not have complex multiplication.

Modular form 17689.2.a.f

sage: E.q_eigenform(10)
 
\(q - 2q^{3} - 2q^{4} - 3q^{5} + q^{9} + 3q^{11} + 4q^{12} - 4q^{13} + 6q^{15} + 4q^{16} + 3q^{17} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.