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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 111540bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111540.bj4 | 111540bi1 | \([0, 1, 0, 36955, -10034400]\) | \(72268906496/606436875\) | \(-46834479458910000\) | \([2]\) | \(663552\) | \(1.8801\) | \(\Gamma_0(N)\)-optimal |
| 111540.bj3 | 111540bi2 | \([0, 1, 0, -533420, -138254700]\) | \(13584145739344/1195803675\) | \(1477610480825107200\) | \([2]\) | \(1327104\) | \(2.2267\) | |
| 111540.bj2 | 111540bi3 | \([0, 1, 0, -2640005, -1653219372]\) | \(-26348629355659264/24169921875\) | \(-1866617542968750000\) | \([2]\) | \(1990656\) | \(2.4295\) | |
| 111540.bj1 | 111540bi4 | \([0, 1, 0, -42249380, -105714969372]\) | \(6749703004355978704/5671875\) | \(7008526668000000\) | \([2]\) | \(3981312\) | \(2.7760\) |
Rank
sage: E.rank()
The elliptic curves in class 111540bi have rank \(0\).
Complex multiplication
The elliptic curves in class 111540bi do not have complex multiplication.Modular form 111540.2.a.bi
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
