Show commands:
SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 100800cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.du4 | 100800cz1 | \([0, 0, 0, 7800, 2189000]\) | \(4499456/180075\) | \(-2100394800000000\) | \([2]\) | \(393216\) | \(1.6199\) | \(\Gamma_0(N)\)-optimal |
| 100800.du3 | 100800cz2 | \([0, 0, 0, -212700, 36146000]\) | \(5702413264/275625\) | \(51438240000000000\) | \([2, 2]\) | \(786432\) | \(1.9665\) | |
| 100800.du2 | 100800cz3 | \([0, 0, 0, -590700, -127906000]\) | \(30534944836/8203125\) | \(6123600000000000000\) | \([2]\) | \(1572864\) | \(2.3131\) | |
| 100800.du1 | 100800cz4 | \([0, 0, 0, -3362700, 2373446000]\) | \(5633270409316/14175\) | \(10581580800000000\) | \([2]\) | \(1572864\) | \(2.3131\) |
Rank
sage: E.rank()
The elliptic curves in class 100800cz have rank \(0\).
Complex multiplication
The elliptic curves in class 100800cz do not have complex multiplication.Modular form 100800.2.a.cz
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.
