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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 34650.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.q1 | 34650q8 | \([1, -1, 0, -57696192, -158249839784]\) | \(1864737106103260904761/129177711985836360\) | \(1471414875588667288125000\) | \([2]\) | \(5308416\) | \(3.3853\) | |
| 34650.q2 | 34650q5 | \([1, -1, 0, -56700567, -164320502909]\) | \(1769857772964702379561/691787250\) | \(7879889144531250\) | \([2]\) | \(1769472\) | \(2.8360\) | |
| 34650.q3 | 34650q6 | \([1, -1, 0, -11391192, 11828425216]\) | \(14351050585434661561/3001282273281600\) | \(34186480894098225000000\) | \([2, 2]\) | \(2654208\) | \(3.0387\) | |
| 34650.q4 | 34650q3 | \([1, -1, 0, -10743192, 13555345216]\) | \(12038605770121350841/757333463040\) | \(8626501477440000000\) | \([2]\) | \(1327104\) | \(2.6922\) | |
| 34650.q5 | 34650q2 | \([1, -1, 0, -3544317, -2566034159]\) | \(432288716775559561/270140062500\) | \(3077064149414062500\) | \([2, 2]\) | \(884736\) | \(2.4894\) | |
| 34650.q6 | 34650q4 | \([1, -1, 0, -2876067, -3563731409]\) | \(-230979395175477481/348191894531250\) | \(-3966123298645019531250\) | \([2]\) | \(1769472\) | \(2.8360\) | |
| 34650.q7 | 34650q1 | \([1, -1, 0, -263817, -23646659]\) | \(178272935636041/81841914000\) | \(932230551656250000\) | \([2]\) | \(442368\) | \(2.1429\) | \(\Gamma_0(N)\)-optimal |
| 34650.q8 | 34650q7 | \([1, -1, 0, 24545808, 71376034216]\) | \(143584693754978072519/276341298967965000\) | \(-3147700108556976328125000\) | \([2]\) | \(5308416\) | \(3.3853\) |
Rank
sage: E.rank()
The elliptic curves in class 34650.q have rank \(1\).
Complex multiplication
The elliptic curves in class 34650.q do not have complex multiplication.Modular form 34650.2.a.q
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
