Properties

Label 77616.fz
Number of curves $4$
Conductor $77616$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("fz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 77616.fz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77616.fz1 77616cj4 \([0, 0, 0, -282342459, 1294787288522]\) \(14171198121996897746/4077720290568771\) \(716247555778716410690131968\) \([2]\) \(35389440\) \(3.8594\)  
77616.fz2 77616cj2 \([0, 0, 0, -258863619, 1602881322770]\) \(21843440425782779332/3100814593569\) \(272327515781739188069376\) \([2, 2]\) \(17694720\) \(3.5128\)  
77616.fz3 77616cj1 \([0, 0, 0, -258854799, 1602996023342]\) \(87364831012240243408/1760913\) \(38662829421689088\) \([2]\) \(8847360\) \(3.1663\) \(\Gamma_0(N)\)-optimal
77616.fz4 77616cj3 \([0, 0, 0, -235525899, 1903634520410]\) \(-8226100326647904626/4152140742401883\) \(-729319434899189202955966464\) \([2]\) \(35389440\) \(3.8594\)  

Rank

sage: E.rank()
 

The elliptic curves in class 77616.fz have rank \(0\).

Complex multiplication

The elliptic curves in class 77616.fz do not have complex multiplication.

Modular form 77616.2.a.fz

sage: E.q_eigenform(10)
 
\(q + 2q^{5} + q^{11} + 6q^{13} - 2q^{17} - 8q^{19} + O(q^{20})\)  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.