Show commands:
SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 142800ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 142800.ex5 | 142800ce1 | \([0, 1, 0, 13992, -1572012]\) | \(4733169839/19518975\) | \(-1249214400000000\) | \([2]\) | \(786432\) | \(1.5794\) | \(\Gamma_0(N)\)-optimal |
| 142800.ex4 | 142800ce2 | \([0, 1, 0, -148008, -19392012]\) | \(5602762882081/716900625\) | \(45881640000000000\) | \([2, 2]\) | \(1572864\) | \(1.9260\) | |
| 142800.ex3 | 142800ce3 | \([0, 1, 0, -598008, 157907988]\) | \(369543396484081/45120132225\) | \(2887688462400000000\) | \([2, 2]\) | \(3145728\) | \(2.2725\) | |
| 142800.ex2 | 142800ce4 | \([0, 1, 0, -2290008, -1334580012]\) | \(20751759537944401/418359375\) | \(26775000000000000\) | \([2]\) | \(3145728\) | \(2.2725\) | |
| 142800.ex1 | 142800ce5 | \([0, 1, 0, -9268008, 10856687988]\) | \(1375634265228629281/24990412335\) | \(1599386389440000000\) | \([2]\) | \(6291456\) | \(2.6191\) | |
| 142800.ex6 | 142800ce6 | \([0, 1, 0, 871992, 813527988]\) | \(1145725929069119/5127181719135\) | \(-328139630024640000000\) | \([2]\) | \(6291456\) | \(2.6191\) |
Rank
sage: E.rank()
The elliptic curves in class 142800ce have rank \(1\).
Complex multiplication
The elliptic curves in class 142800ce do not have complex multiplication.Modular form 142800.2.a.ce
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
