Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 61200bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
61200.m4 | 61200bp1 | \([0, 0, 0, -4553175, 113517481750]\) | \(-3579968623693264/1906997690433375\) | \(-5560805265303721500000000\) | \([2]\) | \(10321920\) | \(3.4272\) | \(\Gamma_0(N)\)-optimal |
61200.m3 | 61200bp2 | \([0, 0, 0, -380397675, 2825987238250]\) | \(521902963282042184836/6241849278890625\) | \(72804929988980250000000000\) | \([2, 2]\) | \(20643840\) | \(3.7738\) | |
61200.m2 | 61200bp3 | \([0, 0, 0, -705522675, -2720320136750]\) | \(1664865424893526702418/826424127435466125\) | \(19278822044814553764000000000\) | \([2]\) | \(41287680\) | \(4.1204\) | |
61200.m1 | 61200bp4 | \([0, 0, 0, -6068784675, 181970359029250]\) | \(1059623036730633329075378/154307373046875\) | \(3599682398437500000000000\) | \([2]\) | \(41287680\) | \(4.1204\) |
Rank
sage: E.rank()
The elliptic curves in class 61200bp have rank \(0\).
Complex multiplication
The elliptic curves in class 61200bp do not have complex multiplication.Modular form 61200.2.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 Cremona 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 Cremona labels.