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Properties

Label 78650.k
Number of curves $3$
Conductor $78650$
CM no
Rank $1$
Graph

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

Elliptic curves in class 78650.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78650.k1 78650p3 \([1, 1, 0, -1390050, 630224500]\) \(-10730978619193/6656\) \(-184242344000000\) \([]\) \(699840\) \(2.0581\)  
78650.k2 78650p2 \([1, 1, 0, -13675, 1221125]\) \(-10218313/17576\) \(-486514939625000\) \([]\) \(233280\) \(1.5087\)  
78650.k3 78650p1 \([1, 1, 0, 1450, -34250]\) \(12167/26\) \(-719696656250\) \([]\) \(77760\) \(0.95944\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 78650.k have rank \(1\).

Complex multiplication

The elliptic curves in class 78650.k do not have complex multiplication.

Modular form 78650.2.a.k

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.