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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 69360do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69360.dn4 | 69360do1 | \([0, 1, 0, 18400, 110580]\) | \(6967871/4080\) | \(-403379329105920\) | \([2]\) | \(221184\) | \(1.4920\) | \(\Gamma_0(N)\)-optimal |
| 69360.dn3 | 69360do2 | \([0, 1, 0, -74080, 813428]\) | \(454756609/260100\) | \(25715432230502400\) | \([2, 2]\) | \(442368\) | \(1.8386\) | |
| 69360.dn2 | 69360do3 | \([0, 1, 0, -767680, -258038092]\) | \(506071034209/2505630\) | \(247725330487173120\) | \([2]\) | \(884736\) | \(2.1851\) | |
| 69360.dn1 | 69360do4 | \([0, 1, 0, -860160, 306126900]\) | \(711882749089/1721250\) | \(170175654466560000\) | \([4]\) | \(884736\) | \(2.1851\) |
Rank
sage: E.rank()
The elliptic curves in class 69360do have rank \(0\).
Complex multiplication
The elliptic curves in class 69360do do not have complex multiplication.Modular form 69360.2.a.do
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
