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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 40656co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40656.cd5 | 40656co1 | \([0, 1, 0, -7784, 581172]\) | \(-7189057/16128\) | \(-117029829869568\) | \([2]\) | \(122880\) | \(1.3880\) | \(\Gamma_0(N)\)-optimal |
| 40656.cd4 | 40656co2 | \([0, 1, 0, -162664, 25176116]\) | \(65597103937/63504\) | \(460804955111424\) | \([2, 2]\) | \(245760\) | \(1.7346\) | |
| 40656.cd3 | 40656co3 | \([0, 1, 0, -201384, 12243636]\) | \(124475734657/63011844\) | \(457233716709310464\) | \([2, 2]\) | \(491520\) | \(2.0812\) | |
| 40656.cd1 | 40656co4 | \([0, 1, 0, -2602024, 1614663092]\) | \(268498407453697/252\) | \(1828591091712\) | \([2]\) | \(491520\) | \(2.0812\) | |
| 40656.cd6 | 40656co5 | \([0, 1, 0, 747256, 95344500]\) | \(6359387729183/4218578658\) | \(-30611330768671285248\) | \([2]\) | \(983040\) | \(2.4278\) | |
| 40656.cd2 | 40656co6 | \([0, 1, 0, -1769544, -897916428]\) | \(84448510979617/933897762\) | \(6776655270487990272\) | \([2]\) | \(983040\) | \(2.4278\) |
Rank
sage: E.rank()
The elliptic curves in class 40656co have rank \(2\).
Complex multiplication
The elliptic curves in class 40656co do not have complex multiplication.Modular form 40656.2.a.co
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
