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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 88200ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 88200.ey4 | 88200ca1 | \([0, 0, 0, -80850, 57409625]\) | \(-2725888/64827\) | \(-1389990081516750000\) | \([2]\) | \(1179648\) | \(2.1620\) | \(\Gamma_0(N)\)-optimal |
| 88200.ey3 | 88200ca2 | \([0, 0, 0, -2781975, 1778026250]\) | \(6940769488/35721\) | \(12254606432964000000\) | \([2, 2]\) | \(2359296\) | \(2.5086\) | |
| 88200.ey2 | 88200ca3 | \([0, 0, 0, -4325475, -415287250]\) | \(6522128932/3720087\) | \(5104918908360432000000\) | \([2]\) | \(4718592\) | \(2.8552\) | |
| 88200.ey1 | 88200ca4 | \([0, 0, 0, -44456475, 114090803750]\) | \(7080974546692/189\) | \(259356749904000000\) | \([2]\) | \(4718592\) | \(2.8552\) |
Rank
sage: E.rank()
The elliptic curves in class 88200ca have rank \(1\).
Complex multiplication
The elliptic curves in class 88200ca do not have complex multiplication.Modular form 88200.2.a.ca
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.
