Properties

Label 66240.ez
Number of curves $6$
Conductor $66240$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ez1")
 
E.isogeny_class()
 

Elliptic curves in class 66240.ez

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
66240.ez1 66240cw4 \([0, 0, 0, -63590412, -195180115984]\) \(148809678420065817601/20700\) \(3955831603200\) \([2]\) \(2359296\) \(2.7420\)  
66240.ez2 66240cw6 \([0, 0, 0, -14878092, 18920874224]\) \(1905890658841300321/293666194803750\) \(56120483777592360960000\) \([4]\) \(4718592\) \(3.0886\)  
66240.ez3 66240cw3 \([0, 0, 0, -4078092, -2882165776]\) \(39248884582600321/3935264062500\) \(752040673689600000000\) \([2, 2]\) \(2359296\) \(2.7420\)  
66240.ez4 66240cw2 \([0, 0, 0, -3974412, -3049671184]\) \(36330796409313601/428490000\) \(81885714186240000\) \([2, 2]\) \(1179648\) \(2.3955\)  
66240.ez5 66240cw1 \([0, 0, 0, -241932, -50250256]\) \(-8194759433281/965779200\) \(-184563279278899200\) \([2]\) \(589824\) \(2.0489\) \(\Gamma_0(N)\)-optimal
66240.ez6 66240cw5 \([0, 0, 0, 5063028, -13964859664]\) \(75108181893694559/484313964843750\) \(-92553840000000000000000\) \([2]\) \(4718592\) \(3.0886\)  

Rank

sage: E.rank()
 

The elliptic curves in class 66240.ez have rank \(0\).

Complex multiplication

The elliptic curves in class 66240.ez do not have complex multiplication.

Modular form 66240.2.a.ez

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.