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Properties

Label 650.b
Number of curves $2$
Conductor $650$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 650.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
650.b1 650g1 \([1, 0, 1, -26, 48]\) \(-2941225/52\) \(-32500\) \([3]\) \(72\) \(-0.33716\) \(\Gamma_0(N)\)-optimal
650.b2 650g2 \([1, 0, 1, 99, 248]\) \(174196775/140608\) \(-87880000\) \([]\) \(216\) \(0.21214\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 650.2.a.b

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