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Properties

Label 61710k
Number of curves $2$
Conductor $61710$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 61710k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.j2 61710k1 \([1, 1, 0, -150042, 18808596]\) \(158428241531/26438400\) \(62340364233734400\) \([2]\) \(1182720\) \(1.9438\) \(\Gamma_0(N)\)-optimal
61710.j1 61710k2 \([1, 1, 0, -682442, -199368924]\) \(14906915211131/1365212880\) \(3219100558119460080\) \([2]\) \(2365440\) \(2.2904\)  

Rank

sage: E.rank()
 

The elliptic curves in class 61710k have rank \(0\).

Complex multiplication

The elliptic curves in class 61710k do not have complex multiplication.

Modular form 61710.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} - q^{12} - 6 q^{13} + 4 q^{14} - q^{15} + q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.