Properties

Label 388416.fa
Number of curves $4$
Conductor $388416$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("fa1")
 
E.isogeny_class()
 

Elliptic curves in class 388416.fa

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388416.fa1 388416fa4 \([0, 1, 0, -629249, -174821409]\) \(17418812548/1753941\) \(2774523701456338944\) \([2]\) \(5898240\) \(2.2751\)  
388416.fa2 388416fa2 \([0, 1, 0, -143729, 17930031]\) \(830321872/127449\) \(50402247171784704\) \([2, 2]\) \(2949120\) \(1.9285\)  
388416.fa3 388416fa1 \([0, 1, 0, -137949, 19674435]\) \(11745974272/357\) \(8823922824192\) \([2]\) \(1474560\) \(1.5819\) \(\Gamma_0(N)\)-optimal
388416.fa4 388416fa3 \([0, 1, 0, 249311, 99132095]\) \(1083360092/3306177\) \(-5229974353589895168\) \([2]\) \(5898240\) \(2.2751\)  

Rank

sage: E.rank()
 

The elliptic curves in class 388416.fa have rank \(2\).

Complex multiplication

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

Modular form 388416.2.a.fa

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