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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1386.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1386.e1 | 1386e2 | \([1, -1, 0, -67005, -6657467]\) | \(45637459887836881/13417633152\) | \(9781454567808\) | \([2]\) | \(10752\) | \(1.4716\) | |
1386.e2 | 1386e1 | \([1, -1, 0, -3645, -131387]\) | \(-7347774183121/6119866368\) | \(-4461382582272\) | \([2]\) | \(5376\) | \(1.1251\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1386.e have rank \(0\).
Complex multiplication
The elliptic curves in class 1386.e do not have complex multiplication.Modular form 1386.2.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.