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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 8976.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8976.w1 | 8976bf5 | \([0, 1, 0, -6319104, 6111977652]\) | \(6812873765474836663297/74052\) | \(303316992\) | \([4]\) | \(98304\) | \(2.1311\) | |
| 8976.w2 | 8976bf3 | \([0, 1, 0, -394944, 95400756]\) | \(1663303207415737537/5483698704\) | \(22461229891584\) | \([2, 4]\) | \(49152\) | \(1.7845\) | |
| 8976.w3 | 8976bf6 | \([0, 1, 0, -389504, 98162100]\) | \(-1595514095015181697/95635786040388\) | \(-391724179621429248\) | \([4]\) | \(98304\) | \(2.1311\) | |
| 8976.w4 | 8976bf2 | \([0, 1, 0, -25024, 1441076]\) | \(423108074414017/23284318464\) | \(95372568428544\) | \([2, 2]\) | \(24576\) | \(1.4379\) | |
| 8976.w5 | 8976bf1 | \([0, 1, 0, -4544, -90828]\) | \(2533811507137/625016832\) | \(2560068943872\) | \([2]\) | \(12288\) | \(1.0914\) | \(\Gamma_0(N)\)-optimal |
| 8976.w6 | 8976bf4 | \([0, 1, 0, 17216, 5850932]\) | \(137763859017023/3683199928848\) | \(-15086386908561408\) | \([2]\) | \(49152\) | \(1.7845\) |
Rank
sage: E.rank()
The elliptic curves in class 8976.w have rank \(1\).
Complex multiplication
The elliptic curves in class 8976.w do not have complex multiplication.Modular form 8976.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
