Show commands:
SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 100800dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.gq4 | 100800dz1 | \([0, 0, 0, 9825, 204500]\) | \(143877824/108045\) | \(-78764805000000\) | \([2]\) | \(196608\) | \(1.3546\) | \(\Gamma_0(N)\)-optimal |
| 100800.gq3 | 100800dz2 | \([0, 0, 0, -45300, 1748000]\) | \(220348864/99225\) | \(4629441600000000\) | \([2, 2]\) | \(393216\) | \(1.7012\) | |
| 100800.gq2 | 100800dz3 | \([0, 0, 0, -360300, -82042000]\) | \(13858588808/229635\) | \(85710804480000000\) | \([2]\) | \(786432\) | \(2.0478\) | |
| 100800.gq1 | 100800dz4 | \([0, 0, 0, -612300, 184322000]\) | \(68017239368/39375\) | \(14696640000000000\) | \([2]\) | \(786432\) | \(2.0478\) |
Rank
sage: E.rank()
The elliptic curves in class 100800dz have rank \(0\).
Complex multiplication
The elliptic curves in class 100800dz do not have complex multiplication.Modular form 100800.2.a.dz
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.
