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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 122400dc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122400.z1 | 122400dc1 | \([0, 0, 0, -1125, 13500]\) | \(216000/17\) | \(12393000000\) | \([2]\) | \(73728\) | \(0.68055\) | \(\Gamma_0(N)\)-optimal |
122400.z2 | 122400dc2 | \([0, 0, 0, 1125, 60750]\) | \(27000/289\) | \(-1685448000000\) | \([2]\) | \(147456\) | \(1.0271\) |
Rank
sage: E.rank()
The elliptic curves in class 122400dc have rank \(1\).
Complex multiplication
The elliptic curves in class 122400dc do not have complex multiplication.Modular form 122400.2.a.dc
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.