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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 6720bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.cm4 | 6720bc1 | \([0, 1, 0, 35, -637]\) | \(4499456/180075\) | \(-184396800\) | \([2]\) | \(2048\) | \(0.26590\) | \(\Gamma_0(N)\)-optimal |
| 6720.cm3 | 6720bc2 | \([0, 1, 0, -945, -11025]\) | \(5702413264/275625\) | \(4515840000\) | \([2, 2]\) | \(4096\) | \(0.61247\) | |
| 6720.cm1 | 6720bc3 | \([0, 1, 0, -14945, -708225]\) | \(5633270409316/14175\) | \(928972800\) | \([2]\) | \(8192\) | \(0.95905\) | |
| 6720.cm2 | 6720bc4 | \([0, 1, 0, -2625, 37023]\) | \(30534944836/8203125\) | \(537600000000\) | \([2]\) | \(8192\) | \(0.95905\) |
Rank
sage: E.rank()
The elliptic curves in class 6720bc have rank \(0\).
Complex multiplication
The elliptic curves in class 6720bc do not have complex multiplication.Modular form 6720.2.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 Cremona 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 Cremona labels.
