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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 436800.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
436800.bg1 | 436800bg4 | \([0, -1, 0, -3130433, -2130367263]\) | \(828279937799497/193444524\) | \(792348770304000000\) | \([2]\) | \(9437184\) | \(2.4251\) | |
436800.bg2 | 436800bg2 | \([0, -1, 0, -218433, -24991263]\) | \(281397674377/96589584\) | \(395630936064000000\) | \([2, 2]\) | \(4718592\) | \(2.0785\) | |
436800.bg3 | 436800bg1 | \([0, -1, 0, -90433, 10208737]\) | \(19968681097/628992\) | \(2576351232000000\) | \([2]\) | \(2359296\) | \(1.7319\) | \(\Gamma_0(N)\)-optimal* |
436800.bg4 | 436800bg3 | \([0, -1, 0, 645567, -174463263]\) | \(7264187703863/7406095788\) | \(-30335368347648000000\) | \([2]\) | \(9437184\) | \(2.4251\) |
Rank
sage: E.rank()
The elliptic curves in class 436800.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 436800.bg do not have complex multiplication.Modular form 436800.2.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.