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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 474320ie
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
474320.ie1 | 474320ie1 | \([0, -1, 0, -179846, 29768411]\) | \(-3937024/55\) | \(-8987173029437680\) | \([]\) | \(4354560\) | \(1.8672\) | \(\Gamma_0(N)\)-optimal |
474320.ie2 | 474320ie2 | \([0, -1, 0, 650214, 148799015]\) | \(186050816/166375\) | \(-27186198414048982000\) | \([]\) | \(13063680\) | \(2.4165\) |
Rank
sage: E.rank()
The elliptic curves in class 474320ie have rank \(1\).
Complex multiplication
The elliptic curves in class 474320ie do not have complex multiplication.Modular form 474320.2.a.ie
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.