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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 43758.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43758.e1 | 43758f1 | \([1, -1, 0, -759042, 254708500]\) | \(66342819962001390625/4812668669952\) | \(3508435460395008\) | \([2]\) | \(473088\) | \(2.0340\) | \(\Gamma_0(N)\)-optimal |
43758.e2 | 43758f2 | \([1, -1, 0, -710082, 288951124]\) | \(-54315282059491182625/17983956399469632\) | \(-13110304215213361728\) | \([2]\) | \(946176\) | \(2.3806\) |
Rank
sage: E.rank()
The elliptic curves in class 43758.e have rank \(1\).
Complex multiplication
The elliptic curves in class 43758.e do not have complex multiplication.Modular form 43758.2.a.e
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.