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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1386.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1386.j1 | 1386j2 | \([1, -1, 1, -4070120, -3114601045]\) | \(10228636028672744397625/167006381634183168\) | \(121747652211319529472\) | \([2]\) | \(66560\) | \(2.6532\) | |
1386.j2 | 1386j1 | \([1, -1, 1, -15080, -136579669]\) | \(-520203426765625/11054534935707648\) | \(-8058755968130875392\) | \([2]\) | \(33280\) | \(2.3067\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1386.j have rank \(0\).
Complex multiplication
The elliptic curves in class 1386.j do not have complex multiplication.Modular form 1386.2.a.j
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.