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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 28224.es
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28224.es1 | 28224fw6 | \([0, 0, 0, -22128204, 40065204368]\) | \(53297461115137/147\) | \(3305011881443328\) | \([2]\) | \(786432\) | \(2.6362\) | |
| 28224.es2 | 28224fw4 | \([0, 0, 0, -1383564, 625494800]\) | \(13027640977/21609\) | \(485836746572169216\) | \([2, 2]\) | \(393216\) | \(2.2896\) | |
| 28224.es3 | 28224fw3 | \([0, 0, 0, -1101324, -442162672]\) | \(6570725617/45927\) | \(1032580140673794048\) | \([2]\) | \(393216\) | \(2.2896\) | |
| 28224.es4 | 28224fw5 | \([0, 0, 0, -960204, 1015494032]\) | \(-4354703137/17294403\) | \(-388831342839926095872\) | \([2]\) | \(786432\) | \(2.6362\) | |
| 28224.es5 | 28224fw2 | \([0, 0, 0, -113484, 3155600]\) | \(7189057/3969\) | \(89235320798969856\) | \([2, 2]\) | \(196608\) | \(1.9430\) | |
| 28224.es6 | 28224fw1 | \([0, 0, 0, 27636, 389648]\) | \(103823/63\) | \(-1416433663475712\) | \([2]\) | \(98304\) | \(1.5965\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 28224.es have rank \(0\).
Complex multiplication
The elliptic curves in class 28224.es do not have complex multiplication.Modular form 28224.2.a.es
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
