Properties

Label 388416.fi
Number of curves $6$
Conductor $388416$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("388416.fi1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 388416.fi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
388416.fi1 388416fi5 [0, 1, 0, -253739299969, 49195904458658687] [2] 1698693120  
388416.fi2 388416fi3 [0, 1, 0, -15888878209, 765610271321471] [2, 2] 849346560  
388416.fi3 388416fi6 [0, 1, 0, -5412003969, 1760186229049215] [2] 1698693120  
388416.fi4 388416fi2 [0, 1, 0, -1678031489, -6649771983489] [2, 2] 424673280  
388416.fi5 388416fi1 [0, 1, 0, -1299233409, -18003032277633] [2] 212336640 \(\Gamma_0(N)\)-optimal
388416.fi6 388416fi4 [0, 1, 0, 6472045951, -52295095693953] [2] 849346560  

Rank

sage: E.rank()
 

The elliptic curves in class 388416.fi have rank \(0\).

Modular form 388416.2.a.fi

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.