Properties

Label 73920.k
Number of curves $4$
Conductor $73920$
CM no
Rank $2$
Graph

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

Elliptic curves in class 73920.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
73920.k1 73920eb4 \([0, -1, 0, -59841, 4067361]\) \(723231880398728/202569142545\) \(6637785662914560\) \([2]\) \(491520\) \(1.7431\)  
73920.k2 73920eb2 \([0, -1, 0, -22041, -1201959]\) \(289119478354624/13074779025\) \(53554294886400\) \([2, 2]\) \(245760\) \(1.3965\)  
73920.k3 73920eb1 \([0, -1, 0, -21796, -1231310]\) \(17893449053367616/39220335\) \(2510101440\) \([2]\) \(122880\) \(1.0499\) \(\Gamma_0(N)\)-optimal
73920.k4 73920eb3 \([0, -1, 0, 11839, -4596735]\) \(5599924283512/281331579375\) \(-9218673192960000\) \([2]\) \(491520\) \(1.7431\)  

Rank

sage: E.rank()
 

The elliptic curves in class 73920.k have rank \(2\).

Complex multiplication

The elliptic curves in class 73920.k do not have complex multiplication.

Modular form 73920.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} - q^{11} + 2 q^{13} + q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.