Show commands:
SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 4200.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4200.u1 | 4200w3 | \([0, 1, 0, -93408, -11019312]\) | \(5633270409316/14175\) | \(226800000000\) | \([2]\) | \(12288\) | \(1.4172\) | |
| 4200.u2 | 4200w4 | \([0, 1, 0, -16408, 586688]\) | \(30534944836/8203125\) | \(131250000000000\) | \([2]\) | \(12288\) | \(1.4172\) | |
| 4200.u3 | 4200w2 | \([0, 1, 0, -5908, -169312]\) | \(5702413264/275625\) | \(1102500000000\) | \([2, 2]\) | \(6144\) | \(1.0706\) | |
| 4200.u4 | 4200w1 | \([0, 1, 0, 217, -10062]\) | \(4499456/180075\) | \(-45018750000\) | \([2]\) | \(3072\) | \(0.72405\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4200.u have rank \(1\).
Complex multiplication
The elliptic curves in class 4200.u do not have complex multiplication.Modular form 4200.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
