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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)
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.