Properties

Label 650.f
Number of curves $2$
Conductor $650$
CM no
Rank $1$
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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 650.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
650.f1 650b2 \([1, 1, 0, -11330, -468940]\) \(-6434774386429585/140608\) \(-3515200\) \([]\) \(1080\) \(0.78239\)  
650.f2 650b1 \([1, 1, 0, -130, -780]\) \(-9836106385/3407872\) \(-85196800\) \([]\) \(360\) \(0.23308\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 650.f have rank \(1\).

Complex multiplication

The elliptic curves in class 650.f do not have complex multiplication.

Modular form 650.2.a.f

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