Properties

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

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Show commands: SageMath
E = EllipticCurve("ee1")
 
E.isogeny_class()
 

Elliptic curves in class 73920ee

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
73920.f3 73920ee1 \([0, -1, 0, -316, 2266]\) \(54698902336/144375\) \(9240000\) \([2]\) \(20480\) \(0.21150\) \(\Gamma_0(N)\)-optimal
73920.f2 73920ee2 \([0, -1, 0, -441, 441]\) \(2320940224/1334025\) \(5464166400\) \([2, 2]\) \(40960\) \(0.55808\)  
73920.f4 73920ee3 \([0, -1, 0, 1759, 1761]\) \(18357958072/10696455\) \(-350501437440\) \([2]\) \(81920\) \(0.90465\)  
73920.f1 73920ee4 \([0, -1, 0, -4641, -119679]\) \(337444269128/1537305\) \(50374410240\) \([2]\) \(81920\) \(0.90465\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 73920.2.a.ee

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