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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 162240bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.fh3 | 162240bm1 | \([0, 1, 0, -1082691796, 13711783649630]\) | \(454357982636417669333824/3003024375\) | \(927681605150040000\) | \([2]\) | \(36126720\) | \(3.5066\) | \(\Gamma_0(N)\)-optimal |
| 162240.fh2 | 162240bm2 | \([0, 1, 0, -1082712921, 13711221796455]\) | \(7099759044484031233216/577161945398025\) | \(11410843535379233391513600\) | \([2, 2]\) | \(72253440\) | \(3.8532\) | |
| 162240.fh4 | 162240bm3 | \([0, 1, 0, -1008792321, 15663804877215]\) | \(-717825640026599866952/254764560814329735\) | \(-40294805504644025347788472320\) | \([2]\) | \(144506880\) | \(4.1998\) | |
| 162240.fh1 | 162240bm4 | \([0, 1, 0, -1156971521, 11722680450495]\) | \(1082883335268084577352/251301565117746585\) | \(39747081215161967450431979520\) | \([2]\) | \(144506880\) | \(4.1998\) |
Rank
sage: E.rank()
The elliptic curves in class 162240bm have rank \(1\).
Complex multiplication
The elliptic curves in class 162240bm do not have complex multiplication.Modular form 162240.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
