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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 115920.ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 115920.ck1 | 115920eq4 | \([0, 0, 0, -24023667, -45321642574]\) | \(513516182162686336369/1944885031250\) | \(5807395585152000000\) | \([2]\) | \(8626176\) | \(2.8151\) | |
| 115920.ck2 | 115920eq3 | \([0, 0, 0, -1523667, -686142574]\) | \(131010595463836369/7704101562500\) | \(23004324000000000000\) | \([2]\) | \(4313088\) | \(2.4685\) | |
| 115920.ck3 | 115920eq2 | \([0, 0, 0, -409107, -10786606]\) | \(2535986675931409/1450751712200\) | \(4331921400601804800\) | \([2]\) | \(2875392\) | \(2.2658\) | |
| 115920.ck4 | 115920eq1 | \([0, 0, 0, -265107, 52314194]\) | \(690080604747409/3406760000\) | \(10172530851840000\) | \([2]\) | \(1437696\) | \(1.9192\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 115920.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 115920.ck do not have complex multiplication.Modular form 115920.2.a.ck
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
