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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 489762.dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.dp1 | 489762dp2 | \([1, -1, 1, -264686, -52282915]\) | \(582810602977/829472\) | \(2918699424924192\) | \([2]\) | \(4147200\) | \(1.8704\) | |
489762.dp2 | 489762dp1 | \([1, -1, 1, -21326, -301219]\) | \(304821217/164864\) | \(580114171413504\) | \([2]\) | \(2073600\) | \(1.5239\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 489762.dp have rank \(0\).
Complex multiplication
The elliptic curves in class 489762.dp do not have complex multiplication.Modular form 489762.2.a.dp
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.