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SageMath
E = EllipticCurve("cq1")
E.isogeny_class()
Elliptic curves in class 31680.cq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.cq1 | 31680dh6 | \([0, 0, 0, -98544972, -376530196336]\) | \(553808571467029327441/12529687500\) | \(2394460569600000000\) | \([2]\) | \(2359296\) | \(3.0508\) | |
| 31680.cq2 | 31680dh4 | \([0, 0, 0, -6811212, 6827041424]\) | \(182864522286982801/463015182960\) | \(88483579396840488960\) | \([2]\) | \(1179648\) | \(2.7042\) | |
| 31680.cq3 | 31680dh3 | \([0, 0, 0, -6166092, -5869178224]\) | \(135670761487282321/643043610000\) | \(122887547568783360000\) | \([2, 2]\) | \(1179648\) | \(2.7042\) | |
| 31680.cq4 | 31680dh5 | \([0, 0, 0, -2998092, -11892179824]\) | \(-15595206456730321/310672490129100\) | \(-59370437425001634201600\) | \([2]\) | \(2359296\) | \(3.0508\) | |
| 31680.cq5 | 31680dh2 | \([0, 0, 0, -590412, 16509584]\) | \(119102750067601/68309049600\) | \(13054062666291609600\) | \([2, 2]\) | \(589824\) | \(2.3576\) | |
| 31680.cq6 | 31680dh1 | \([0, 0, 0, 146868, 2058896]\) | \(1833318007919/1070530560\) | \(-204581575914946560\) | \([2]\) | \(294912\) | \(2.0111\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31680.cq have rank \(0\).
Complex multiplication
The elliptic curves in class 31680.cq do not have complex multiplication.Modular form 31680.2.a.cq
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
