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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 221880bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221880.e4 | 221880bd1 | \([0, -1, 0, -1664716, -1053713804]\) | \(-315278049616/114259815\) | \(-184903109766803439360\) | \([2]\) | \(7805952\) | \(2.5994\) | \(\Gamma_0(N)\)-optimal |
| 221880.e3 | 221880bd2 | \([0, -1, 0, -28623136, -58928049860]\) | \(400649568576484/33698025\) | \(218129868860700902400\) | \([2, 2]\) | \(15611904\) | \(2.9460\) | |
| 221880.e2 | 221880bd3 | \([0, -1, 0, -30620056, -50231862644]\) | \(245245463376482/57692266875\) | \(746892828747103645440000\) | \([2]\) | \(31223808\) | \(3.2926\) | |
| 221880.e1 | 221880bd4 | \([0, -1, 0, -457960936, -3772013079380]\) | \(820480625548035842/5805\) | \(75152409598863360\) | \([2]\) | \(31223808\) | \(3.2926\) |
Rank
sage: E.rank()
The elliptic curves in class 221880bd have rank \(0\).
Complex multiplication
The elliptic curves in class 221880bd do not have complex multiplication.Modular form 221880.2.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 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.
