Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4056.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4056.i1 | 4056a5 | \([0, -1, 0, -64952, -6349812]\) | \(3065617154/9\) | \(88967743488\) | \([2]\) | \(9216\) | \(1.3303\) | |
| 4056.i2 | 4056a4 | \([0, -1, 0, -10872, 439932]\) | \(28756228/3\) | \(14827957248\) | \([2]\) | \(4608\) | \(0.98370\) | |
| 4056.i3 | 4056a3 | \([0, -1, 0, -4112, -95460]\) | \(1556068/81\) | \(400354845696\) | \([2, 2]\) | \(4608\) | \(0.98370\) | |
| 4056.i4 | 4056a2 | \([0, -1, 0, -732, 5940]\) | \(35152/9\) | \(11120967936\) | \([2, 2]\) | \(2304\) | \(0.63712\) | |
| 4056.i5 | 4056a1 | \([0, -1, 0, 113, 532]\) | \(2048/3\) | \(-231686832\) | \([2]\) | \(1152\) | \(0.29055\) | \(\Gamma_0(N)\)-optimal |
| 4056.i6 | 4056a6 | \([0, -1, 0, 2648, -384788]\) | \(207646/6561\) | \(-64857485002752\) | \([2]\) | \(9216\) | \(1.3303\) |
Rank
sage: E.rank()
The elliptic curves in class 4056.i have rank \(1\).
Complex multiplication
The elliptic curves in class 4056.i do not have complex multiplication.Modular form 4056.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
