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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 129360.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.w1 | 129360du6 | \([0, -1, 0, -10388016, -12882777984]\) | \(257260669489908001/14267882475\) | \(6875554223314022400\) | \([2]\) | \(6291456\) | \(2.6800\) | |
| 129360.w2 | 129360du4 | \([0, -1, 0, -686016, -177038784]\) | \(74093292126001/14707625625\) | \(7087462183549440000\) | \([2, 2]\) | \(3145728\) | \(2.3334\) | |
| 129360.w3 | 129360du2 | \([0, -1, 0, -211696, 35077120]\) | \(2177286259681/161417025\) | \(77785299248025600\) | \([2, 2]\) | \(1572864\) | \(1.9868\) | |
| 129360.w4 | 129360du1 | \([0, -1, 0, -207776, 36522816]\) | \(2058561081361/12705\) | \(6122416312320\) | \([2]\) | \(786432\) | \(1.6402\) | \(\Gamma_0(N)\)-optimal |
| 129360.w5 | 129360du3 | \([0, -1, 0, 199904, 154605760]\) | \(1833318007919/22507682505\) | \(-10846233964669931520\) | \([2]\) | \(3145728\) | \(2.3334\) | |
| 129360.w6 | 129360du5 | \([0, -1, 0, 1426864, -1054306560]\) | \(666688497209279/1381398046875\) | \(-665682324753600000000\) | \([2]\) | \(6291456\) | \(2.6800\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.w have rank \(1\).
Complex multiplication
The elliptic curves in class 129360.w do not have complex multiplication.Modular form 129360.2.a.w
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
