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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 46389k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 46389.d6 | 46389k1 | \([1, 0, 0, 2163, 8712]\) | \(103823/63\) | \(-679090565727\) | \([2]\) | \(52992\) | \(0.95956\) | \(\Gamma_0(N)\)-optimal |
| 46389.d5 | 46389k2 | \([1, 0, 0, -8882, 68355]\) | \(7189057/3969\) | \(42782705640801\) | \([2, 2]\) | \(105984\) | \(1.3061\) | |
| 46389.d3 | 46389k3 | \([1, 0, 0, -86197, -9688798]\) | \(6570725617/45927\) | \(495057022414983\) | \([2]\) | \(211968\) | \(1.6527\) | |
| 46389.d2 | 46389k4 | \([1, 0, 0, -108287, 13686840]\) | \(13027640977/21609\) | \(232928064044361\) | \([2, 2]\) | \(211968\) | \(1.6527\) | |
| 46389.d4 | 46389k5 | \([1, 0, 0, -75152, 22229043]\) | \(-4354703137/17294403\) | \(-186420093923503587\) | \([2]\) | \(423936\) | \(1.9993\) | |
| 46389.d1 | 46389k6 | \([1, 0, 0, -1731902, 877125297]\) | \(53297461115137/147\) | \(1584544653363\) | \([2]\) | \(423936\) | \(1.9993\) |
Rank
sage: E.rank()
The elliptic curves in class 46389k have rank \(1\).
Complex multiplication
The elliptic curves in class 46389k do not have complex multiplication.Modular form 46389.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
