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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 12144.bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12144.bn1 | 12144bd4 | \([0, 1, 0, -345432, 78028308]\) | \(1112891236915770073/327888\) | \(1343029248\) | \([4]\) | \(36864\) | \(1.5559\) | |
| 12144.bn2 | 12144bd3 | \([0, 1, 0, -25432, 748052]\) | \(444142553850073/196663299888\) | \(805532876341248\) | \([2]\) | \(36864\) | \(1.5559\) | |
| 12144.bn3 | 12144bd2 | \([0, 1, 0, -21592, 1213460]\) | \(271808161065433/147476736\) | \(604064710656\) | \([2, 2]\) | \(18432\) | \(1.2093\) | |
| 12144.bn4 | 12144bd1 | \([0, 1, 0, -1112, 25620]\) | \(-37159393753/49741824\) | \(-203742511104\) | \([2]\) | \(9216\) | \(0.86274\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12144.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 12144.bn do not have complex multiplication.Modular form 12144.2.a.bn
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.
