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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 56784cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56784.h3 | 56784cb1 | \([0, -1, 0, -18984, 908400]\) | \(38272753/4368\) | \(86358023012352\) | \([2]\) | \(193536\) | \(1.4064\) | \(\Gamma_0(N)\)-optimal |
| 56784.h2 | 56784cb2 | \([0, -1, 0, -73064, -6619536]\) | \(2181825073/298116\) | \(5893935070593024\) | \([2, 2]\) | \(387072\) | \(1.7530\) | |
| 56784.h4 | 56784cb3 | \([0, -1, 0, 116216, -35390096]\) | \(8780064047/32388174\) | \(-640333945883713536\) | \([2]\) | \(774144\) | \(2.0996\) | |
| 56784.h1 | 56784cb4 | \([0, -1, 0, -1127624, -460502160]\) | \(8020417344913/187278\) | \(3702600236654592\) | \([2]\) | \(774144\) | \(2.0996\) |
Rank
sage: E.rank()
The elliptic curves in class 56784cb have rank \(1\).
Complex multiplication
The elliptic curves in class 56784cb do not have complex multiplication.Modular form 56784.2.a.cb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
