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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 72450.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.k1 | 72450l2 | \([1, -1, 0, -349242, 78707916]\) | \(89332607016927/1060723384\) | \(55936584703125000\) | \([2]\) | \(737280\) | \(2.0252\) | |
72450.k2 | 72450l1 | \([1, -1, 0, -4242, 3152916]\) | \(-160103007/81288256\) | \(-4286685375000000\) | \([2]\) | \(368640\) | \(1.6786\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.k have rank \(2\).
Complex multiplication
The elliptic curves in class 72450.k do not have complex multiplication.Modular form 72450.2.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.