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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 283920.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
283920.bp1 | 283920bp7 | \([0, -1, 0, -108674990376, 13789346035420656]\) | \(7179471593960193209684686321/49441793310\) | \(977494396620176547840\) | \([2]\) | \(445906944\) | \(4.5634\) | |
283920.bp2 | 283920bp6 | \([0, -1, 0, -6792191176, 215459944884976]\) | \(1752803993935029634719121/4599740941532100\) | \(90939682710550995226214400\) | \([2, 2]\) | \(222953472\) | \(4.2168\) | |
283920.bp3 | 283920bp8 | \([0, -1, 0, -6708718696, 221013602702320]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-1777407187254745459656821760000\) | \([2]\) | \(445906944\) | \(4.5634\) | |
283920.bp4 | 283920bp4 | \([0, -1, 0, -1342279176, 18897685672176]\) | \(13527956825588849127121/25701087819771000\) | \(508126175224877222866944000\) | \([2]\) | \(148635648\) | \(4.0141\) | |
283920.bp5 | 283920bp3 | \([0, -1, 0, -429733256, 3279608202480]\) | \(443915739051786565201/21894701746029840\) | \(432871601930455228336373760\) | \([2]\) | \(111476736\) | \(3.8703\) | |
283920.bp6 | 283920bp2 | \([0, -1, 0, -111959176, 80679464176]\) | \(7850236389974007121/4400862921000000\) | \(87007742995861868544000000\) | \([2, 2]\) | \(74317824\) | \(3.6675\) | |
283920.bp7 | 283920bp1 | \([0, -1, 0, -69560456, -222081315600]\) | \(1882742462388824401/11650189824000\) | \(230331355521808859136000\) | \([2]\) | \(37158912\) | \(3.3209\) | \(\Gamma_0(N)\)-optimal |
283920.bp8 | 283920bp5 | \([0, -1, 0, 439981304, 639684782320]\) | \(476437916651992691759/284661685546875000\) | \(-5627935071243576000000000000\) | \([2]\) | \(148635648\) | \(4.0141\) |
Rank
sage: E.rank()
The elliptic curves in class 283920.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 283920.bp do not have complex multiplication.Modular form 283920.2.a.bp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.