Show commands:
SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 285600cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 285600.cr3 | 285600cr1 | \([0, -1, 0, -2058, -30888]\) | \(964430272/127449\) | \(127449000000\) | \([2, 2]\) | \(229376\) | \(0.85886\) | \(\Gamma_0(N)\)-optimal |
| 285600.cr2 | 285600cr2 | \([0, -1, 0, -8433, 268737]\) | \(1036433728/122451\) | \(7836864000000\) | \([2]\) | \(458752\) | \(1.2054\) | |
| 285600.cr4 | 285600cr3 | \([0, -1, 0, 3192, -167388]\) | \(449455096/1753941\) | \(-14031528000000\) | \([2]\) | \(458752\) | \(1.2054\) | |
| 285600.cr1 | 285600cr4 | \([0, -1, 0, -31808, -2172888]\) | \(444893916104/9639\) | \(77112000000\) | \([2]\) | \(458752\) | \(1.2054\) |
Rank
sage: E.rank()
The elliptic curves in class 285600cr have rank \(1\).
Complex multiplication
The elliptic curves in class 285600cr do not have complex multiplication.Modular form 285600.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
