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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 340032cv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 340032.cv6 | 340032cv1 | \([0, -1, 0, 6669823, 849132033]\) | \(125177609053596564863/73635189229502208\) | \(-19303023045378626813952\) | \([2]\) | \(19660800\) | \(2.9656\) | \(\Gamma_0(N)\)-optimal |
| 340032.cv5 | 340032cv2 | \([0, -1, 0, -26922497, 6848720385]\) | \(8232463578739844255617/4687062591766850064\) | \(1228685336056129143177216\) | \([2, 2]\) | \(39321600\) | \(3.3122\) | |
| 340032.cv2 | 340032cv3 | \([0, -1, 0, -315463937, 2152500576513]\) | \(13244420128496241770842177/29965867631164664892\) | \(7855372404304029913448448\) | \([2]\) | \(78643200\) | \(3.6587\) | |
| 340032.cv3 | 340032cv4 | \([0, -1, 0, -275858177, -1755366958335]\) | \(8856076866003496152467137/46664863048067576004\) | \(12232913858872626643992576\) | \([2, 2]\) | \(78643200\) | \(3.6587\) | |
| 340032.cv4 | 340032cv5 | \([0, -1, 0, -126553217, -3648225380223]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-5620140096955553345715044352\) | \([4]\) | \(157286400\) | \(4.0053\) | |
| 340032.cv1 | 340032cv6 | \([0, -1, 0, -4408134017, -112648294945407]\) | \(36136672427711016379227705697/1011258101510224722\) | \(265095243762296349523968\) | \([2]\) | \(157286400\) | \(4.0053\) |
Rank
sage: E.rank()
The elliptic curves in class 340032cv have rank \(1\).
Complex multiplication
The elliptic curves in class 340032cv do not have complex multiplication.Modular form 340032.2.a.cv
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 Cremona 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 Cremona labels.
