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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 3630n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3630.n3 | 3630n1 | \([1, 1, 1, -168616, 26579609]\) | \(299270638153369/1069200\) | \(1894153021200\) | \([4]\) | \(19200\) | \(1.5738\) | \(\Gamma_0(N)\)-optimal |
| 3630.n2 | 3630n2 | \([1, 1, 1, -171036, 25774233]\) | \(312341975961049/17862322500\) | \(31644193910422500\) | \([2, 2]\) | \(38400\) | \(1.9204\) | |
| 3630.n1 | 3630n3 | \([1, 1, 1, -503786, -105462367]\) | \(7981893677157049/1917731420550\) | \(3397378193120978550\) | \([2]\) | \(76800\) | \(2.2669\) | |
| 3630.n4 | 3630n4 | \([1, 1, 1, 122994, 105515169]\) | \(116149984977671/2779502343750\) | \(-4924057951596093750\) | \([2]\) | \(76800\) | \(2.2669\) |
Rank
sage: E.rank()
The elliptic curves in class 3630n have rank \(1\).
Complex multiplication
The elliptic curves in class 3630n do not have complex multiplication.Modular form 3630.2.a.n
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.
