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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 22050eg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.dt1 | 22050eg1 | \([1, -1, 1, -165605, -9872103]\) | \(1092727/540\) | \(248212514556562500\) | \([2]\) | \(258048\) | \(2.0308\) | \(\Gamma_0(N)\)-optimal |
22050.dt2 | 22050eg2 | \([1, -1, 1, 606145, -76242603]\) | \(53582633/36450\) | \(-16754344732567968750\) | \([2]\) | \(516096\) | \(2.3774\) |
Rank
sage: E.rank()
The elliptic curves in class 22050eg have rank \(0\).
Complex multiplication
The elliptic curves in class 22050eg do not have complex multiplication.Modular form 22050.2.a.eg
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.