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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 21840ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.k3 | 21840ba1 | \([0, -1, 0, -16336, 801280]\) | \(117713838907729/1322517105\) | \(5417030062080\) | \([2]\) | \(55296\) | \(1.2572\) | \(\Gamma_0(N)\)-optimal |
| 21840.k2 | 21840ba2 | \([0, -1, 0, -29856, -702144]\) | \(718576775407009/362361861225\) | \(1484234183577600\) | \([2, 2]\) | \(110592\) | \(1.6038\) | |
| 21840.k4 | 21840ba3 | \([0, -1, 0, 110544, -5531904]\) | \(36472485598112591/24291459037755\) | \(-99497816218644480\) | \([2]\) | \(221184\) | \(1.9503\) | |
| 21840.k1 | 21840ba4 | \([0, -1, 0, -386576, -92307840]\) | \(1559802282754777489/1481059636875\) | \(6066420272640000\) | \([2]\) | \(221184\) | \(1.9503\) |
Rank
sage: E.rank()
The elliptic curves in class 21840ba have rank \(0\).
Complex multiplication
The elliptic curves in class 21840ba do not have complex multiplication.Modular form 21840.2.a.ba
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.
