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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 163254.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163254.u1 | 163254du6 | \([1, 1, 0, -13373649, -18829864503]\) | \(54804145548726848737/637608031452\) | \(3077612184684796668\) | \([2]\) | \(9437184\) | \(2.6984\) | |
| 163254.u2 | 163254du3 | \([1, 1, 0, -2993669, 1992371997]\) | \(614716917569296417/19093020912\) | \(92158365175229808\) | \([2]\) | \(4718592\) | \(2.3519\) | |
| 163254.u3 | 163254du4 | \([1, 1, 0, -857509, -278441795]\) | \(14447092394873377/1439452851984\) | \(6947963981032039056\) | \([2, 2]\) | \(4718592\) | \(2.3519\) | |
| 163254.u4 | 163254du2 | \([1, 1, 0, -195029, 28286445]\) | \(169967019783457/26337394944\) | \(127125574952253696\) | \([2, 2]\) | \(2359296\) | \(2.0053\) | |
| 163254.u5 | 163254du1 | \([1, 1, 0, 21291, 2457837]\) | \(221115865823/664731648\) | \(-3208532701151232\) | \([2]\) | \(1179648\) | \(1.6587\) | \(\Gamma_0(N)\)-optimal |
| 163254.u6 | 163254du5 | \([1, 1, 0, 1058951, -1344376847]\) | \(27207619911317663/177609314617308\) | \(-857286238278653810172\) | \([2]\) | \(9437184\) | \(2.6984\) |
Rank
sage: E.rank()
The elliptic curves in class 163254.u have rank \(1\).
Complex multiplication
The elliptic curves in class 163254.u do not have complex multiplication.Modular form 163254.2.a.u
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
