Properties

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

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

Elliptic curves in class 388416.fi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388416.fi1 388416fi5 \([0, 1, 0, -253739299969, 49195904458658687]\) \(285531136548675601769470657/17941034271597192\) \(113522233222638328114380275712\) \([2]\) \(1698693120\) \(5.0355\)  
388416.fi2 388416fi3 \([0, 1, 0, -15888878209, 765610271321471]\) \(70108386184777836280897/552468975892674624\) \(3495757879959557329741166936064\) \([2, 2]\) \(849346560\) \(4.6889\)  
388416.fi3 388416fi6 \([0, 1, 0, -5412003969, 1760186229049215]\) \(-2770540998624539614657/209924951154647363208\) \(-1328304093480073402804118841458688\) \([2]\) \(1698693120\) \(5.0355\)  
388416.fi4 388416fi2 \([0, 1, 0, -1678031489, -6649771983489]\) \(82582985847542515777/44772582831427584\) \(283299363527537112462995226624\) \([2, 2]\) \(424673280\) \(4.3423\)  
388416.fi5 388416fi1 \([0, 1, 0, -1299233409, -18003032277633]\) \(38331145780597164097/55468445663232\) \(350977637618552186603569152\) \([2]\) \(212336640\) \(3.9957\) \(\Gamma_0(N)\)-optimal
388416.fi6 388416fi4 \([0, 1, 0, 6472045951, -52295095693953]\) \(4738217997934888496063/2928751705237796928\) \(-18531731732966928484293157060608\) \([2]\) \(849346560\) \(4.6889\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 388416.2.a.fi

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.