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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 46800.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
46800.df1 | 46800o4 | \([0, 0, 0, -1308675, 576049250]\) | \(10625310339698/3855735\) | \(89946586080000000\) | \([2]\) | \(589824\) | \(2.2217\) | |
46800.df2 | 46800o3 | \([0, 0, 0, -678675, -210820750]\) | \(1481943889298/34543665\) | \(805834617120000000\) | \([2]\) | \(589824\) | \(2.2217\) | |
46800.df3 | 46800o2 | \([0, 0, 0, -93675, 6214250]\) | \(7793764996/3080025\) | \(35925411600000000\) | \([2, 2]\) | \(294912\) | \(1.8751\) | |
46800.df4 | 46800o1 | \([0, 0, 0, 18825, 701750]\) | \(253012016/219375\) | \(-639697500000000\) | \([2]\) | \(147456\) | \(1.5285\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 46800.df have rank \(0\).
Complex multiplication
The elliptic curves in class 46800.df do not have complex multiplication.Modular form 46800.2.a.df
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.