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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 224400fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 224400.co4 | 224400fd1 | \([0, -1, 0, -2924008, -1923513488]\) | \(43199583152847841/89760000\) | \(5744640000000000\) | \([2]\) | \(3538944\) | \(2.2733\) | \(\Gamma_0(N)\)-optimal |
| 224400.co3 | 224400fd2 | \([0, -1, 0, -2956008, -1879225488]\) | \(44633474953947361/1967006250000\) | \(125888400000000000000\) | \([2, 2]\) | \(7077888\) | \(2.6199\) | |
| 224400.co2 | 224400fd3 | \([0, -1, 0, -7956008, 6160774512]\) | \(870220733067747361/247623269602500\) | \(15847889254560000000000\) | \([2, 2]\) | \(14155776\) | \(2.9665\) | |
| 224400.co5 | 224400fd4 | \([0, -1, 0, 1531992, -7085305488]\) | \(6213165856218719/342407226562500\) | \(-21914062500000000000000\) | \([2]\) | \(14155776\) | \(2.9665\) | |
| 224400.co1 | 224400fd5 | \([0, -1, 0, -116856008, 486191974512]\) | \(2757381641970898311361/379829992662450\) | \(24309119530396800000000\) | \([2]\) | \(28311552\) | \(3.3130\) | |
| 224400.co6 | 224400fd6 | \([0, -1, 0, 20943992, 40609574512]\) | \(15875306080318016639/20322604533582450\) | \(-1300646690149276800000000\) | \([2]\) | \(28311552\) | \(3.3130\) |
Rank
sage: E.rank()
The elliptic curves in class 224400fd have rank \(1\).
Complex multiplication
The elliptic curves in class 224400fd do not have complex multiplication.Modular form 224400.2.a.fd
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.
