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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 19152.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19152.bc1 | 19152bg3 | \([0, 0, 0, -2434995, -1042395534]\) | \(19804628171203875/5638671302656\) | \(454598521856729284608\) | \([2]\) | \(663552\) | \(2.6705\) | |
| 19152.bc2 | 19152bg1 | \([0, 0, 0, -2235075, -1286134078]\) | \(11165451838341046875/572244736\) | \(63285689843712\) | \([2]\) | \(221184\) | \(2.1212\) | \(\Gamma_0(N)\)-optimal |
| 19152.bc3 | 19152bg2 | \([0, 0, 0, -2231235, -1290773566]\) | \(-11108001800138902875/79947274872976\) | \(-8841529022752161792\) | \([2]\) | \(442368\) | \(2.4678\) | |
| 19152.bc4 | 19152bg4 | \([0, 0, 0, 6412365, -6876344718]\) | \(361682234074684125/462672528510976\) | \(-37301384719079590330368\) | \([2]\) | \(1327104\) | \(3.0171\) |
Rank
sage: E.rank()
The elliptic curves in class 19152.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 19152.bc do not have complex multiplication.Modular form 19152.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
