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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 6720.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.bg1 | 6720bu3 | \([0, -1, 0, -89185, 6377665]\) | \(2394165105226952/854262178245\) | \(27992463056732160\) | \([2]\) | \(61440\) | \(1.8568\) | |
| 6720.bg2 | 6720bu2 | \([0, -1, 0, -79385, 8633625]\) | \(13507798771700416/3544416225\) | \(14517928857600\) | \([2, 2]\) | \(30720\) | \(1.5103\) | |
| 6720.bg3 | 6720bu1 | \([0, -1, 0, -79380, 8634762]\) | \(864335783029582144/59535\) | \(3810240\) | \([2]\) | \(15360\) | \(1.1637\) | \(\Gamma_0(N)\)-optimal |
| 6720.bg4 | 6720bu4 | \([0, -1, 0, -69665, 10816737]\) | \(-1141100604753992/875529151875\) | \(-28689339248640000\) | \([4]\) | \(61440\) | \(1.8568\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 6720.bg do not have complex multiplication.Modular form 6720.2.a.bg
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.
