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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 121968.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.bc1 | 121968ei4 | \([0, 0, 0, -1603086771, -24704929507214]\) | \(86129359107301290313/9166294368\) | \(48488347937575315832832\) | \([2]\) | \(44236800\) | \(3.7807\) | |
121968.bc2 | 121968ei2 | \([0, 0, 0, -100441011, -384006823310]\) | \(21184262604460873/216872764416\) | \(1147225000311795675561984\) | \([2, 2]\) | \(22118400\) | \(3.4341\) | |
121968.bc3 | 121968ei3 | \([0, 0, 0, -25169331, -946211001230]\) | \(-333345918055753/72923718045024\) | \(-385755734161554619745304576\) | \([2]\) | \(44236800\) | \(3.7807\) | |
121968.bc4 | 121968ei1 | \([0, 0, 0, -11230131, 4792033906]\) | \(29609739866953/15259926528\) | \(80722765087561243164672\) | \([2]\) | \(11059200\) | \(3.0876\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 121968.bc do not have complex multiplication.Modular form 121968.2.a.bc
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.