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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 193200.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
193200.df1 | 193200ef5 | \([0, -1, 0, -31653608, 68555993712]\) | \(54804145548726848737/637608031452\) | \(40806914012928000000\) | \([2]\) | \(12582912\) | \(2.9138\) | |
193200.df2 | 193200ef4 | \([0, -1, 0, -7085608, -7257062288]\) | \(614716917569296417/19093020912\) | \(1221953338368000000\) | \([2]\) | \(6291456\) | \(2.5673\) | |
193200.df3 | 193200ef3 | \([0, -1, 0, -2029608, 1013273712]\) | \(14447092394873377/1439452851984\) | \(92124982526976000000\) | \([2, 2]\) | \(6291456\) | \(2.5673\) | |
193200.df4 | 193200ef2 | \([0, -1, 0, -461608, -103142288]\) | \(169967019783457/26337394944\) | \(1685593276416000000\) | \([2, 2]\) | \(3145728\) | \(2.2207\) | |
193200.df5 | 193200ef1 | \([0, -1, 0, 50392, -8934288]\) | \(221115865823/664731648\) | \(-42542825472000000\) | \([2]\) | \(1572864\) | \(1.8741\) | \(\Gamma_0(N)\)-optimal |
193200.df6 | 193200ef6 | \([0, -1, 0, 2506392, 4896089712]\) | \(27207619911317663/177609314617308\) | \(-11366996135507712000000\) | \([2]\) | \(12582912\) | \(2.9138\) |
Rank
sage: E.rank()
The elliptic curves in class 193200.df have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.df do not have complex multiplication.Modular form 193200.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.