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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1386.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1386.d1 | 1386a1 | \([1, -1, 0, -231, -451]\) | \(69426531/34496\) | \(678984768\) | \([2]\) | \(576\) | \(0.38764\) | \(\Gamma_0(N)\)-optimal |
1386.d2 | 1386a2 | \([1, -1, 0, 849, -4123]\) | \(3436115229/2324168\) | \(-45746598744\) | \([2]\) | \(1152\) | \(0.73421\) |
Rank
sage: E.rank()
The elliptic curves in class 1386.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1386.d do not have complex multiplication.Modular form 1386.2.a.d
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.