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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 61200.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
61200.j1 | 61200ck1 | \([0, 0, 0, -5250, -115625]\) | \(702464/153\) | \(3485531250000\) | \([2]\) | \(112640\) | \(1.1204\) | \(\Gamma_0(N)\)-optimal |
61200.j2 | 61200ck2 | \([0, 0, 0, 11625, -706250]\) | \(476656/867\) | \(-316021500000000\) | \([2]\) | \(225280\) | \(1.4670\) |
Rank
sage: E.rank()
The elliptic curves in class 61200.j have rank \(1\).
Complex multiplication
The elliptic curves in class 61200.j do not have complex multiplication.Modular form 61200.2.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.