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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 36414.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 36414.k1 | 36414bi4 | \([1, -1, 0, -3495798, -2514876984]\) | \(268498407453697/252\) | \(4434264525852\) | \([2]\) | \(655360\) | \(2.1550\) | |
| 36414.k2 | 36414bi6 | \([1, -1, 0, -2377368, 1397937366]\) | \(84448510979617/933897762\) | \(16433133796861801362\) | \([2]\) | \(1310720\) | \(2.5016\) | |
| 36414.k3 | 36414bi3 | \([1, -1, 0, -270558, -19103040]\) | \(124475734657/63011844\) | \(1108774541895715044\) | \([2, 2]\) | \(655360\) | \(2.1550\) | |
| 36414.k4 | 36414bi2 | \([1, -1, 0, -218538, -39234780]\) | \(65597103937/63504\) | \(1117434660514704\) | \([2, 2]\) | \(327680\) | \(1.8084\) | |
| 36414.k5 | 36414bi1 | \([1, -1, 0, -10458, -906444]\) | \(-7189057/16128\) | \(-283792929654528\) | \([2]\) | \(163840\) | \(1.4619\) | \(\Gamma_0(N)\)-optimal |
| 36414.k6 | 36414bi5 | \([1, -1, 0, 1003932, -148336326]\) | \(6359387729183/4218578658\) | \(-74231324177324351058\) | \([2]\) | \(1310720\) | \(2.5016\) |
Rank
sage: E.rank()
The elliptic curves in class 36414.k have rank \(1\).
Complex multiplication
The elliptic curves in class 36414.k do not have complex multiplication.Modular form 36414.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
