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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 55488cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55488.be2 | 55488cd1 | \([0, -1, 0, -4726113, -3951571455]\) | \(1845026709625/793152\) | \(5018684261004214272\) | \([2]\) | \(1327104\) | \(2.5484\) | \(\Gamma_0(N)\)-optimal |
| 55488.be3 | 55488cd2 | \([0, -1, 0, -3986273, -5231346687]\) | \(-1107111813625/1228691592\) | \(-7774569255828153434112\) | \([2]\) | \(2654208\) | \(2.8950\) | |
| 55488.be1 | 55488cd3 | \([0, -1, 0, -13881633, 15055121601]\) | \(46753267515625/11591221248\) | \(73343671380965094064128\) | \([2]\) | \(3981312\) | \(3.0978\) | |
| 55488.be4 | 55488cd4 | \([0, -1, 0, 33468127, 95426604225]\) | \(655215969476375/1001033261568\) | \(-6334056870023698367643648\) | \([2]\) | \(7962624\) | \(3.4443\) |
Rank
sage: E.rank()
The elliptic curves in class 55488cd have rank \(0\).
Complex multiplication
The elliptic curves in class 55488cd do not have complex multiplication.Modular form 55488.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
