Show commands:
SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 85008bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85008.f6 | 85008bm1 | \([0, -1, 0, 1667456, -106975232]\) | \(125177609053596564863/73635189229502208\) | \(-301609735084041043968\) | \([2]\) | \(2457600\) | \(2.6190\) | \(\Gamma_0(N)\)-optimal |
85008.f5 | 85008bm2 | \([0, -1, 0, -6730624, -852724736]\) | \(8232463578739844255617/4687062591766850064\) | \(19198208375877017862144\) | \([2, 2]\) | \(4915200\) | \(2.9656\) | |
85008.f3 | 85008bm3 | \([0, -1, 0, -68964544, 219455352064]\) | \(8856076866003496152467137/46664863048067576004\) | \(191139279044884791312384\) | \([2, 4]\) | \(9830400\) | \(3.3122\) | |
85008.f2 | 85008bm4 | \([0, -1, 0, -78865984, -269023139072]\) | \(13244420128496241770842177/29965867631164664892\) | \(122740193817250467397632\) | \([2]\) | \(9830400\) | \(3.3122\) | |
85008.f4 | 85008bm5 | \([0, -1, 0, -31638304, 456043991680]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-87814689014930521026797568\) | \([4]\) | \(19660800\) | \(3.6587\) | |
85008.f1 | 85008bm6 | \([0, -1, 0, -1102033504, 14081587884928]\) | \(36136672427711016379227705697/1011258101510224722\) | \(4142113183785880461312\) | \([8]\) | \(19660800\) | \(3.6587\) |
Rank
sage: E.rank()
The elliptic curves in class 85008bm have rank \(0\).
Complex multiplication
The elliptic curves in class 85008bm do not have complex multiplication.Modular form 85008.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.