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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 235200.cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.cj1 | 235200cj6 | \([0, -1, 0, -51353633, -141625900863]\) | \(62161150998242/1607445\) | \(387306079856640000000\) | \([2]\) | \(18874368\) | \(3.0572\) | |
| 235200.cj2 | 235200cj4 | \([0, -1, 0, -3333633, -2031760863]\) | \(34008619684/4862025\) | \(585740676326400000000\) | \([2, 2]\) | \(9437184\) | \(2.7106\) | |
| 235200.cj3 | 235200cj2 | \([0, -1, 0, -883633, 288389137]\) | \(2533446736/275625\) | \(8301313440000000000\) | \([2, 2]\) | \(4718592\) | \(2.3640\) | |
| 235200.cj4 | 235200cj1 | \([0, -1, 0, -859133, 306788637]\) | \(37256083456/525\) | \(988251600000000\) | \([2]\) | \(2359296\) | \(2.0174\) | \(\Gamma_0(N)\)-optimal |
| 235200.cj5 | 235200cj3 | \([0, -1, 0, 1174367, 1430579137]\) | \(1486779836/8203125\) | \(-988251600000000000000\) | \([2]\) | \(9437184\) | \(2.7106\) | |
| 235200.cj6 | 235200cj5 | \([0, -1, 0, 5486367, -10966420863]\) | \(75798394558/259416045\) | \(-62505038393763840000000\) | \([2]\) | \(18874368\) | \(3.0572\) |
Rank
sage: E.rank()
The elliptic curves in class 235200.cj have rank \(1\).
Complex multiplication
The elliptic curves in class 235200.cj do not have complex multiplication.Modular form 235200.2.a.cj
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
