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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 266805df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
266805.df3 | 266805df1 | \([1, -1, 0, -10860075, -13762511064]\) | \(932288503609/779625\) | \(118456156298050349625\) | \([2]\) | \(13271040\) | \(2.7797\) | \(\Gamma_0(N)\)-optimal |
266805.df2 | 266805df2 | \([1, -1, 0, -13261320, -7225841925]\) | \(1697509118089/833765625\) | \(126682278263192735015625\) | \([2, 2]\) | \(26542080\) | \(3.1262\) | |
266805.df1 | 266805df3 | \([1, -1, 0, -113313195, 459276030450]\) | \(1058993490188089/13182390375\) | \(2002931274193252602420375\) | \([2]\) | \(53084160\) | \(3.4728\) | |
266805.df4 | 266805df4 | \([1, -1, 0, 48370635, -55409704344]\) | \(82375335041831/56396484375\) | \(-8568877047023318115234375\) | \([2]\) | \(53084160\) | \(3.4728\) |
Rank
sage: E.rank()
The elliptic curves in class 266805df have rank \(0\).
Complex multiplication
The elliptic curves in class 266805df do not have complex multiplication.Modular form 266805.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.