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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1200k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1200.c2 | 1200k1 | \([0, -1, 0, 27, -243]\) | \(20480/243\) | \(-24883200\) | \([]\) | \(240\) | \(0.10005\) | \(\Gamma_0(N)\)-optimal |
1200.c1 | 1200k2 | \([0, -1, 0, -3333, 77037]\) | \(-102400/3\) | \(-120000000000\) | \([]\) | \(1200\) | \(0.90477\) |
Rank
sage: E.rank()
The elliptic curves in class 1200k have rank \(0\).
Complex multiplication
The elliptic curves in class 1200k do not have complex multiplication.Modular form 1200.2.a.k
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.