Properties

Label 61009.b
Number of curves $3$
Conductor $61009$
CM no
Rank $2$
Graph

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

Elliptic curves in class 61009.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61009.b1 61009b3 \([0, -1, 1, -46936257, 123784336522]\) \(-50357871050752/19\) \(-4314548154650851\) \([]\) \(2332800\) \(2.7881\)  
61009.b2 61009b2 \([0, -1, 1, -569417, 176136937]\) \(-89915392/6859\) \(-1557551883828957211\) \([]\) \(777600\) \(2.2388\)  
61009.b3 61009b1 \([0, -1, 1, 40673, 125972]\) \(32768/19\) \(-4314548154650851\) \([]\) \(259200\) \(1.6895\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 61009.b have rank \(2\).

Complex multiplication

The elliptic curves in class 61009.b do not have complex multiplication.

Modular form 61009.2.a.b

sage: E.q_eigenform(10)
 
\(q + 2q^{3} - 2q^{4} - 3q^{5} + q^{7} + q^{9} - 3q^{11} - 4q^{12} - 6q^{15} + 4q^{16} - 3q^{17} + O(q^{20})\)  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.