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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 12480bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12480.m5 | 12480bm1 | \([0, -1, 0, -37181, -2974275]\) | \(-5551350318708736/550618236675\) | \(-563833074355200\) | \([2]\) | \(61440\) | \(1.5710\) | \(\Gamma_0(N)\)-optimal |
| 12480.m4 | 12480bm2 | \([0, -1, 0, -608401, -182451599]\) | \(1520107298839022416/13013105625\) | \(213206722560000\) | \([2, 2]\) | \(122880\) | \(1.9176\) | |
| 12480.m1 | 12480bm3 | \([0, -1, 0, -9734401, -11686687199]\) | \(1556580279686303289604/114075\) | \(7476019200\) | \([2]\) | \(245760\) | \(2.2641\) | |
| 12480.m3 | 12480bm4 | \([0, -1, 0, -621921, -173904255]\) | \(405929061432816484/35083409765625\) | \(2299226342400000000\) | \([2, 2]\) | \(245760\) | \(2.2641\) | |
| 12480.m2 | 12480bm5 | \([0, -1, 0, -2138241, 1005489441]\) | \(8248670337458940482/1446075439453125\) | \(189540000000000000000\) | \([2]\) | \(491520\) | \(2.6107\) | |
| 12480.m6 | 12480bm6 | \([0, -1, 0, 678079, -806484255]\) | \(263059523447441758/2294739983908125\) | \(-300776159170805760000\) | \([2]\) | \(491520\) | \(2.6107\) |
Rank
sage: E.rank()
The elliptic curves in class 12480bm have rank \(0\).
Complex multiplication
The elliptic curves in class 12480bm do not have complex multiplication.Modular form 12480.2.a.bm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
