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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 97104.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
97104.bc1 | 97104bi4 | \([0, -1, 0, -23485392, -43799289408]\) | \(14489843500598257/6246072\) | \(617533414928252928\) | \([2]\) | \(5308416\) | \(2.7563\) | |
97104.bc2 | 97104bi3 | \([0, -1, 0, -3139792, 1132969408]\) | \(34623662831857/14438442312\) | \(1427492444399286386688\) | \([2]\) | \(5308416\) | \(2.7563\) | |
97104.bc3 | 97104bi2 | \([0, -1, 0, -1475152, -676827200]\) | \(3590714269297/73410624\) | \(7257923592736997376\) | \([2, 2]\) | \(2654208\) | \(2.4097\) | |
97104.bc4 | 97104bi1 | \([0, -1, 0, 4528, -31686720]\) | \(103823/4386816\) | \(-433713454654685184\) | \([2]\) | \(1327104\) | \(2.0632\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 97104.bc have rank \(2\).
Complex multiplication
The elliptic curves in class 97104.bc do not have complex multiplication.Modular form 97104.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.