Properties

Label 388416.fe
Number of curves $4$
Conductor $388416$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("fe1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 388416.fe

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388416.fe1 388416fe4 \([0, 1, 0, -93941569, -350488256833]\) \(14489843500598257/6246072\) \(39522138555408187392\) \([2]\) \(42467328\) \(3.1029\)  
388416.fe2 388416fe3 \([0, 1, 0, -12559169, 9051196095]\) \(34623662831857/14438442312\) \(91359516441554328748032\) \([2]\) \(42467328\) \(3.1029\)  
388416.fe3 388416fe2 \([0, 1, 0, -5900609, -5420518209]\) \(3590714269297/73410624\) \(464507109935167832064\) \([2, 2]\) \(21233664\) \(2.7563\)  
388416.fe4 388416fe1 \([0, 1, 0, 18111, -253475649]\) \(103823/4386816\) \(-27757661097899851776\) \([2]\) \(10616832\) \(2.4097\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 388416.fe have rank \(1\).

Complex multiplication

The elliptic curves in class 388416.fe do not have complex multiplication.

Modular form 388416.2.a.fe

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.