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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 204480dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
204480.fe2 | 204480dm1 | \([0, 0, 0, -636492, -168481424]\) | \(149222774347921/22187500000\) | \(4240097280000000000\) | \([]\) | \(4147200\) | \(2.2988\) | \(\Gamma_0(N)\)-optimal |
204480.fe1 | 204480dm2 | \([0, 0, 0, -104460492, 410937867376]\) | \(659648323242974383921/90211467550\) | \(17239679918132428800\) | \([]\) | \(20736000\) | \(3.1035\) |
Rank
sage: E.rank()
The elliptic curves in class 204480dm have rank \(1\).
Complex multiplication
The elliptic curves in class 204480dm do not have complex multiplication.Modular form 204480.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.