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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 162624ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.fu4 | 162624ep1 | \([0, 1, 0, -4991169, 1418198175]\) | \(29609739866953/15259926528\) | \(7086772243626775805952\) | \([2]\) | \(11059200\) | \(2.8848\) | \(\Gamma_0(N)\)-optimal |
| 162624.fu2 | 162624ep2 | \([0, 1, 0, -44640449, -113794679649]\) | \(21184262604460873/216872764416\) | \(100716598106934051078144\) | \([2, 2]\) | \(22118400\) | \(3.2314\) | |
| 162624.fu3 | 162624ep3 | \([0, 1, 0, -11186369, -280362543969]\) | \(-333345918055753/72923718045024\) | \(-33866072683593272515362816\) | \([2]\) | \(44236800\) | \(3.5780\) | |
| 162624.fu1 | 162624ep4 | \([0, 1, 0, -712483009, -7320216607585]\) | \(86129359107301290313/9166294368\) | \(4256864565164362432512\) | \([2]\) | \(44236800\) | \(3.5780\) |
Rank
sage: E.rank()
The elliptic curves in class 162624ep have rank \(0\).
Complex multiplication
The elliptic curves in class 162624ep do not have complex multiplication.Modular form 162624.2.a.ep
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.
