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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 31680bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.dd6 | 31680bu1 | \([0, 0, 0, 146868, -2058896]\) | \(1833318007919/1070530560\) | \(-204581575914946560\) | \([2]\) | \(294912\) | \(2.0111\) | \(\Gamma_0(N)\)-optimal |
| 31680.dd5 | 31680bu2 | \([0, 0, 0, -590412, -16509584]\) | \(119102750067601/68309049600\) | \(13054062666291609600\) | \([2, 2]\) | \(589824\) | \(2.3576\) | |
| 31680.dd3 | 31680bu3 | \([0, 0, 0, -6166092, 5869178224]\) | \(135670761487282321/643043610000\) | \(122887547568783360000\) | \([2, 2]\) | \(1179648\) | \(2.7042\) | |
| 31680.dd2 | 31680bu4 | \([0, 0, 0, -6811212, -6827041424]\) | \(182864522286982801/463015182960\) | \(88483579396840488960\) | \([2]\) | \(1179648\) | \(2.7042\) | |
| 31680.dd4 | 31680bu5 | \([0, 0, 0, -2998092, 11892179824]\) | \(-15595206456730321/310672490129100\) | \(-59370437425001634201600\) | \([2]\) | \(2359296\) | \(3.0508\) | |
| 31680.dd1 | 31680bu6 | \([0, 0, 0, -98544972, 376530196336]\) | \(553808571467029327441/12529687500\) | \(2394460569600000000\) | \([2]\) | \(2359296\) | \(3.0508\) |
Rank
sage: E.rank()
The elliptic curves in class 31680bu have rank \(0\).
Complex multiplication
The elliptic curves in class 31680bu do not have complex multiplication.Modular form 31680.2.a.bu
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
