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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 882e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 882.b5 | 882e1 | \([1, -1, 0, -1773, 63909]\) | \(-7189057/16128\) | \(-1383235999488\) | \([2]\) | \(1536\) | \(1.0182\) | \(\Gamma_0(N)\)-optimal |
| 882.b4 | 882e2 | \([1, -1, 0, -37053, 2752245]\) | \(65597103937/63504\) | \(5446491747984\) | \([2, 2]\) | \(3072\) | \(1.3648\) | |
| 882.b3 | 882e3 | \([1, -1, 0, -45873, 1349865]\) | \(124475734657/63011844\) | \(5404281436937124\) | \([2, 2]\) | \(6144\) | \(1.7114\) | |
| 882.b1 | 882e4 | \([1, -1, 0, -592713, 175784769]\) | \(268498407453697/252\) | \(21613062492\) | \([2]\) | \(6144\) | \(1.7114\) | |
| 882.b2 | 882e5 | \([1, -1, 0, -403083, -97454421]\) | \(84448510979617/933897762\) | \(80096788457321202\) | \([2]\) | \(12288\) | \(2.0579\) | |
| 882.b6 | 882e6 | \([1, -1, 0, 170217, 10295991]\) | \(6359387729183/4218578658\) | \(-361811127630045618\) | \([2]\) | \(12288\) | \(2.0579\) |
Rank
sage: E.rank()
The elliptic curves in class 882e have rank \(1\).
Complex multiplication
The elliptic curves in class 882e do not have complex multiplication.Modular form 882.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
