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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 121968.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.bg1 | 121968bk4 | \([0, 0, 0, -165891, -25973134]\) | \(381775972/567\) | \(749836634545152\) | \([2]\) | \(737280\) | \(1.7552\) | |
| 121968.bg2 | 121968bk2 | \([0, 0, 0, -13431, -146410]\) | \(810448/441\) | \(145801567828224\) | \([2, 2]\) | \(368640\) | \(1.4086\) | |
| 121968.bg3 | 121968bk1 | \([0, 0, 0, -7986, 272855]\) | \(2725888/21\) | \(433933237584\) | \([2]\) | \(184320\) | \(1.0621\) | \(\Gamma_0(N)\)-optimal |
| 121968.bg4 | 121968bk3 | \([0, 0, 0, 51909, -1152646]\) | \(11696828/7203\) | \(-9525702431443968\) | \([2]\) | \(737280\) | \(1.7552\) |
Rank
sage: E.rank()
The elliptic curves in class 121968.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.bg do not have complex multiplication.Modular form 121968.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
