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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 44100dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44100.dm2 | 44100dm1 | \([0, 0, 0, -33883500, -126259371875]\) | \(-1605176213504/1640558367\) | \(-4397010231615137718750000\) | \([2]\) | \(7741440\) | \(3.4241\) | \(\Gamma_0(N)\)-optimal |
44100.dm1 | 44100dm2 | \([0, 0, 0, -636675375, -6181303756250]\) | \(665567485783184/257298363\) | \(11033741267079961500000000\) | \([2]\) | \(15482880\) | \(3.7706\) |
Rank
sage: E.rank()
The elliptic curves in class 44100dm have rank \(1\).
Complex multiplication
The elliptic curves in class 44100dm do not have complex multiplication.Modular form 44100.2.a.dm
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.