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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 277440cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277440.cd3 | 277440cd1 | \([0, -1, 0, -1868481, 859071681]\) | \(114013572049/15667200\) | \(99134503921070899200\) | \([2]\) | \(10616832\) | \(2.5637\) | \(\Gamma_0(N)\)-optimal |
| 277440.cd2 | 277440cd2 | \([0, -1, 0, -7787201, -7496977215]\) | \(8253429989329/936360000\) | \(5924835585907752960000\) | \([2, 2]\) | \(21233664\) | \(2.9103\) | |
| 277440.cd4 | 277440cd3 | \([0, -1, 0, 10708799, -37730538815]\) | \(21464092074671/109596256200\) | \(-693472381152572945203200\) | \([2]\) | \(42467328\) | \(3.2569\) | |
| 277440.cd1 | 277440cd4 | \([0, -1, 0, -120982721, -512145244479]\) | \(30949975477232209/478125000\) | \(3025344968294400000000\) | \([2]\) | \(42467328\) | \(3.2569\) |
Rank
sage: E.rank()
The elliptic curves in class 277440cd have rank \(1\).
Complex multiplication
The elliptic curves in class 277440cd do not have complex multiplication.Modular form 277440.2.a.cd
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.
