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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 19110.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.bd1 | 19110bg3 | \([1, 0, 1, -384333, -91740302]\) | \(53365044437418169/41984670\) | \(4939454440830\) | \([2]\) | \(147456\) | \(1.7423\) | |
| 19110.bd2 | 19110bg4 | \([1, 0, 1, -56033, 3082538]\) | \(165369706597369/60703354530\) | \(7141688957099970\) | \([2]\) | \(147456\) | \(1.7423\) | |
| 19110.bd3 | 19110bg2 | \([1, 0, 1, -24183, -1414682]\) | \(13293525831769/365192100\) | \(42964485372900\) | \([2, 2]\) | \(73728\) | \(1.3957\) | |
| 19110.bd4 | 19110bg1 | \([1, 0, 1, 317, -72082]\) | \(30080231/19110000\) | \(-2248272390000\) | \([2]\) | \(36864\) | \(1.0491\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 19110.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 19110.bd do not have complex multiplication.Modular form 19110.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
