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SageMath
E = EllipticCurve("ho1")
E.isogeny_class()
Elliptic curves in class 286650.ho
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
286650.ho1 | 286650ho3 | \([1, -1, 0, -86474817, 309537043591]\) | \(53365044437418169/41984670\) | \(56263473240079218750\) | \([2]\) | \(28311552\) | \(3.0963\) | |
286650.ho2 | 286650ho4 | \([1, -1, 0, -12607317, -10416173909]\) | \(165369706597369/60703354530\) | \(81348300776966845781250\) | \([2]\) | \(28311552\) | \(3.0963\) | |
286650.ho3 | 286650ho2 | \([1, -1, 0, -5441067, 4769109841]\) | \(13293525831769/365192100\) | \(489392341200689062500\) | \([2, 2]\) | \(14155776\) | \(2.7497\) | |
286650.ho4 | 286650ho1 | \([1, -1, 0, 71433, 243347341]\) | \(30080231/19110000\) | \(-25609227692343750000\) | \([2]\) | \(7077888\) | \(2.4032\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286650.ho have rank \(0\).
Complex multiplication
The elliptic curves in class 286650.ho do not have complex multiplication.Modular form 286650.2.a.ho
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.