Properties

Label 388416ba
Number of curves $6$
Conductor $388416$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ba1")
 
E.isogeny_class()
 

Elliptic curves in class 388416ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
388416.ba5 388416ba1 \([0, -1, 0, -1299233409, 18003032277633]\) \(38331145780597164097/55468445663232\) \(350977637618552186603569152\) \([2]\) \(212336640\) \(3.9957\) \(\Gamma_0(N)\)-optimal
388416.ba4 388416ba2 \([0, -1, 0, -1678031489, 6649771983489]\) \(82582985847542515777/44772582831427584\) \(283299363527537112462995226624\) \([2, 2]\) \(424673280\) \(4.3423\)  
388416.ba6 388416ba3 \([0, -1, 0, 6472045951, 52295095693953]\) \(4738217997934888496063/2928751705237796928\) \(-18531731732966928484293157060608\) \([2]\) \(849346560\) \(4.6889\)  
388416.ba2 388416ba4 \([0, -1, 0, -15888878209, -765610271321471]\) \(70108386184777836280897/552468975892674624\) \(3495757879959557329741166936064\) \([2, 2]\) \(849346560\) \(4.6889\)  
388416.ba3 388416ba5 \([0, -1, 0, -5412003969, -1760186229049215]\) \(-2770540998624539614657/209924951154647363208\) \(-1328304093480073402804118841458688\) \([2]\) \(1698693120\) \(5.0355\)  
388416.ba1 388416ba6 \([0, -1, 0, -253739299969, -49195904458658687]\) \(285531136548675601769470657/17941034271597192\) \(113522233222638328114380275712\) \([2]\) \(1698693120\) \(5.0355\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 388416.2.a.ba

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 Cremona 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 Cremona labels.