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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 72450.ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.ed1 | 72450eh6 | \([1, -1, 1, -19055255, 32020960497]\) | \(67176973097223766561/91487391870\) | \(1042098573019218750\) | \([2]\) | \(3145728\) | \(2.7315\) | |
| 72450.ed2 | 72450eh4 | \([1, -1, 1, -1201505, 491237997]\) | \(16840406336564161/604708416900\) | \(6888006811251562500\) | \([2, 2]\) | \(1572864\) | \(2.3849\) | |
| 72450.ed3 | 72450eh2 | \([1, -1, 1, -189005, -21087003]\) | \(65553197996161/20996010000\) | \(239157676406250000\) | \([2, 2]\) | \(786432\) | \(2.0383\) | |
| 72450.ed4 | 72450eh1 | \([1, -1, 1, -171005, -27171003]\) | \(48551226272641/9273600\) | \(105632100000000\) | \([2]\) | \(393216\) | \(1.6917\) | \(\Gamma_0(N)\)-optimal |
| 72450.ed5 | 72450eh5 | \([1, -1, 1, 452245, 1738165497]\) | \(898045580910239/115117148363070\) | \(-1311256268073094218750\) | \([2]\) | \(3145728\) | \(2.7315\) | |
| 72450.ed6 | 72450eh3 | \([1, -1, 1, 535495, -144252003]\) | \(1490881681033919/1650501562500\) | \(-18800244360351562500\) | \([2]\) | \(1572864\) | \(2.3849\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.ed have rank \(0\).
Complex multiplication
The elliptic curves in class 72450.ed do not have complex multiplication.Modular form 72450.2.a.ed
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
