Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 10320p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10320.c2 | 10320p1 | \([0, -1, 0, -281056, -57254144]\) | \(599437478278595809/33854760000\) | \(138669096960000\) | \([2]\) | \(69120\) | \(1.7780\) | \(\Gamma_0(N)\)-optimal |
| 10320.c3 | 10320p2 | \([0, -1, 0, -265056, -64076544]\) | \(-502780379797811809/143268096832200\) | \(-586826124624691200\) | \([2]\) | \(138240\) | \(2.1246\) | |
| 10320.c1 | 10320p3 | \([0, -1, 0, -548896, 68177920]\) | \(4465136636671380769/2096375976562500\) | \(8586756000000000000\) | \([2]\) | \(207360\) | \(2.3274\) | |
| 10320.c4 | 10320p4 | \([0, -1, 0, 1951104, 514177920]\) | \(200541749524551119231/144008551960031250\) | \(-589859028828288000000\) | \([2]\) | \(414720\) | \(2.6739\) |
Rank
sage: E.rank()
The elliptic curves in class 10320p have rank \(0\).
Complex multiplication
The elliptic curves in class 10320p do not have complex multiplication.Modular form 10320.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
