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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 333270.ed
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.ed1 | 333270ed2 | \([1, -1, 1, -288692, 59775559]\) | \(8099892914322789/12250000\) | \(4024235250000\) | \([2]\) | \(2359296\) | \(1.6866\) | |
333270.ed2 | 333270ed1 | \([1, -1, 1, -18212, 919111]\) | \(2033419614309/76832000\) | \(25240003488000\) | \([2]\) | \(1179648\) | \(1.3400\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.ed have rank \(2\).
Complex multiplication
The elliptic curves in class 333270.ed do not have complex multiplication.Modular form 333270.2.a.ed
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.