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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 42432.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42432.k1 | 42432bw2 | \([0, -1, 0, -531713, 148359969]\) | \(253674278705546500/2058765672717\) | \(134923267127181312\) | \([2]\) | \(393216\) | \(2.1139\) | |
42432.k2 | 42432bw1 | \([0, -1, 0, -530673, 148972113]\) | \(1008754689437602000/67254057\) | \(1101890469888\) | \([2]\) | \(196608\) | \(1.7673\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42432.k have rank \(2\).
Complex multiplication
The elliptic curves in class 42432.k do not have complex multiplication.Modular form 42432.2.a.k
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.