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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 14400.ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 14400.ck1 | 14400ds5 | \([0, 0, 0, -345900, -78302000]\) | \(3065617154/9\) | \(13436928000000\) | \([2]\) | \(65536\) | \(1.7484\) | |
| 14400.ck2 | 14400ds4 | \([0, 0, 0, -57900, 5362000]\) | \(28756228/3\) | \(2239488000000\) | \([2]\) | \(32768\) | \(1.4018\) | |
| 14400.ck3 | 14400ds3 | \([0, 0, 0, -21900, -1190000]\) | \(1556068/81\) | \(60466176000000\) | \([2, 2]\) | \(32768\) | \(1.4018\) | |
| 14400.ck4 | 14400ds2 | \([0, 0, 0, -3900, 70000]\) | \(35152/9\) | \(1679616000000\) | \([2, 2]\) | \(16384\) | \(1.0552\) | |
| 14400.ck5 | 14400ds1 | \([0, 0, 0, 600, 7000]\) | \(2048/3\) | \(-34992000000\) | \([2]\) | \(8192\) | \(0.70867\) | \(\Gamma_0(N)\)-optimal |
| 14400.ck6 | 14400ds6 | \([0, 0, 0, 14100, -4718000]\) | \(207646/6561\) | \(-9795520512000000\) | \([2]\) | \(65536\) | \(1.7484\) |
Rank
sage: E.rank()
The elliptic curves in class 14400.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 14400.ck do not have complex multiplication.Modular form 14400.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
