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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 214774e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 214774.o4 | 214774e1 | \([1, -1, 1, -23814886, -45137954955]\) | \(-10090256344188054273/107965577101312\) | \(-15982780187590764986368\) | \([2]\) | \(20275200\) | \(3.0768\) | \(\Gamma_0(N)\)-optimal |
| 214774.o3 | 214774e2 | \([1, -1, 1, -382011366, -2873743918219]\) | \(41647175116728660507393/4693358285056\) | \(694785466123780374784\) | \([2, 2]\) | \(40550400\) | \(3.4234\) | |
| 214774.o2 | 214774e3 | \([1, -1, 1, -382984726, -2858362494155]\) | \(41966336340198080824833/442001722607124848\) | \(65432117945677124607669872\) | \([4]\) | \(81100800\) | \(3.7700\) | |
| 214774.o1 | 214774e4 | \([1, -1, 1, -6112181686, -183924205348939]\) | \(170586815436843383543017473/2166416\) | \(320707318503824\) | \([2]\) | \(81100800\) | \(3.7700\) |
Rank
sage: E.rank()
The elliptic curves in class 214774e have rank \(1\).
Complex multiplication
The elliptic curves in class 214774e do not have complex multiplication.Modular form 214774.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
