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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 30912.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30912.w1 | 30912bk6 | \([0, -1, 0, -5064577, 4388596513]\) | \(54804145548726848737/637608031452\) | \(167145119796953088\) | \([2]\) | \(786432\) | \(2.4557\) | |
| 30912.w2 | 30912bk4 | \([0, -1, 0, -1133697, -464225247]\) | \(614716917569296417/19093020912\) | \(5005120873955328\) | \([2]\) | \(393216\) | \(2.1091\) | |
| 30912.w3 | 30912bk3 | \([0, -1, 0, -324737, 64914465]\) | \(14447092394873377/1439452851984\) | \(377343928430493696\) | \([2, 2]\) | \(393216\) | \(2.1091\) | |
| 30912.w4 | 30912bk2 | \([0, -1, 0, -73857, -6586335]\) | \(169967019783457/26337394944\) | \(6904190060199936\) | \([2, 2]\) | \(196608\) | \(1.7625\) | |
| 30912.w5 | 30912bk1 | \([0, -1, 0, 8063, -573407]\) | \(221115865823/664731648\) | \(-174255413133312\) | \([2]\) | \(98304\) | \(1.4160\) | \(\Gamma_0(N)\)-optimal |
| 30912.w6 | 30912bk5 | \([0, -1, 0, 401023, 313269537]\) | \(27207619911317663/177609314617308\) | \(-46559216171039588352\) | \([2]\) | \(786432\) | \(2.4557\) |
Rank
sage: E.rank()
The elliptic curves in class 30912.w have rank \(1\).
Complex multiplication
The elliptic curves in class 30912.w do not have complex multiplication.Modular form 30912.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
