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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 21450.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21450.bm1 | 21450bf4 | \([1, 0, 1, -152651, 22943198]\) | \(25176685646263969/57915000\) | \(904921875000\) | \([2]\) | \(110592\) | \(1.5377\) | |
| 21450.bm2 | 21450bf2 | \([1, 0, 1, -9651, 349198]\) | \(6361447449889/294465600\) | \(4601025000000\) | \([2, 2]\) | \(55296\) | \(1.1912\) | |
| 21450.bm3 | 21450bf1 | \([1, 0, 1, -1651, -18802]\) | \(31824875809/8785920\) | \(137280000000\) | \([2]\) | \(27648\) | \(0.84458\) | \(\Gamma_0(N)\)-optimal |
| 21450.bm4 | 21450bf3 | \([1, 0, 1, 5349, 1339198]\) | \(1083523132511/50179392120\) | \(-784053001875000\) | \([2]\) | \(110592\) | \(1.5377\) |
Rank
sage: E.rank()
The elliptic curves in class 21450.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 21450.bm do not have complex multiplication.Modular form 21450.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
