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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 42432bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42432.c2 | 42432bz1 | \([0, -1, 0, -4016064, -3096430506]\) | \(111929798417942466883648/14535930476691\) | \(930299550508224\) | \([2]\) | \(642048\) | \(2.2877\) | \(\Gamma_0(N)\)-optimal |
42432.c1 | 42432bz2 | \([0, -1, 0, -4027049, -3078628215]\) | \(1763293530283953913792/19924870320721197\) | \(81612268833674022912\) | \([2]\) | \(1284096\) | \(2.6343\) |
Rank
sage: E.rank()
The elliptic curves in class 42432bz have rank \(0\).
Complex multiplication
The elliptic curves in class 42432bz do not have complex multiplication.Modular form 42432.2.a.bz
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.