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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 55488dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55488.ch1 | 55488dz1 | \([0, 1, 0, -46625, -2475681]\) | \(1771561/612\) | \(3872441559416832\) | \([2]\) | \(442368\) | \(1.6928\) | \(\Gamma_0(N)\)-optimal |
55488.ch2 | 55488dz2 | \([0, 1, 0, 138335, -17087521]\) | \(46268279/46818\) | \(-296241779295387648\) | \([2]\) | \(884736\) | \(2.0394\) |
Rank
sage: E.rank()
The elliptic curves in class 55488dz have rank \(1\).
Complex multiplication
The elliptic curves in class 55488dz do not have complex multiplication.Modular form 55488.2.a.dz
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.