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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 912.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 912.k1 | 912k3 | \([0, 1, 0, -1400832, 637689780]\) | \(74220219816682217473/16416\) | \(67239936\) | \([2]\) | \(5760\) | \(1.7946\) | |
| 912.k2 | 912k2 | \([0, 1, 0, -87552, 9941940]\) | \(18120364883707393/269485056\) | \(1103810789376\) | \([2, 2]\) | \(2880\) | \(1.4480\) | |
| 912.k3 | 912k4 | \([0, 1, 0, -84992, 10553268]\) | \(-16576888679672833/2216253521952\) | \(-9077774425915392\) | \([4]\) | \(5760\) | \(1.7946\) | |
| 912.k4 | 912k1 | \([0, 1, 0, -5632, 144308]\) | \(4824238966273/537919488\) | \(2203318222848\) | \([2]\) | \(1440\) | \(1.1014\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 912.k have rank \(0\).
Complex multiplication
The elliptic curves in class 912.k do not have complex multiplication.Modular form 912.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
