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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 126960bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 126960.c3 | 126960bm1 | \([0, -1, 0, -87461, -6746964]\) | \(31238127616/9703125\) | \(22982571767250000\) | \([2]\) | \(1064448\) | \(1.8439\) | \(\Gamma_0(N)\)-optimal |
| 126960.c4 | 126960bm2 | \([0, -1, 0, 243164, -45892964]\) | \(41957807024/48205125\) | \(-1826838664635168000\) | \([2]\) | \(2128896\) | \(2.1905\) | |
| 126960.c1 | 126960bm3 | \([0, -1, 0, -6435461, -6281586264]\) | \(12444451776495616/912525\) | \(2161383193755600\) | \([2]\) | \(3193344\) | \(2.3932\) | |
| 126960.c2 | 126960bm4 | \([0, -1, 0, -6422236, -6308702804]\) | \(-772993034343376/6661615005\) | \(-252456473456874097920\) | \([2]\) | \(6386688\) | \(2.7398\) |
Rank
sage: E.rank()
The elliptic curves in class 126960bm have rank \(1\).
Complex multiplication
The elliptic curves in class 126960bm do not have complex multiplication.Modular form 126960.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
