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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 10230.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10230.bf1 | 10230bg3 | \([1, 0, 0, -8731875, -9911311503]\) | \(73628549562506871957390001/178215946908754500240\) | \(178215946908754500240\) | \([2]\) | \(880000\) | \(2.7646\) | |
| 10230.bf2 | 10230bg4 | \([1, 0, 0, -5510855, -17315148075]\) | \(-18508902577171306222471921/118801759721890483665900\) | \(-118801759721890483665900\) | \([2]\) | \(1760000\) | \(3.1111\) | |
| 10230.bf3 | 10230bg1 | \([1, 0, 0, -486675, 130612257]\) | \(12747965531857798561201/2986780262400000\) | \(2986780262400000\) | \([10]\) | \(176000\) | \(1.9599\) | \(\Gamma_0(N)\)-optimal |
| 10230.bf4 | 10230bg2 | \([1, 0, 0, -430355, 162005025]\) | \(-8814635019030000319921/6242069790000000000\) | \(-6242069790000000000\) | \([10]\) | \(352000\) | \(2.3064\) |
Rank
sage: E.rank()
The elliptic curves in class 10230.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 10230.bf do not have complex multiplication.Modular form 10230.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
