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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 162240fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.hj6 | 162240fc1 | \([0, 1, 0, 162015, 10106175]\) | \(371694959/249600\) | \(-315823627016601600\) | \([2]\) | \(2064384\) | \(2.0465\) | \(\Gamma_0(N)\)-optimal |
| 162240.hj5 | 162240fc2 | \([0, 1, 0, -703265, 83308863]\) | \(30400540561/15210000\) | \(19245502271324160000\) | \([2, 2]\) | \(4128768\) | \(2.3931\) | |
| 162240.hj2 | 162240fc3 | \([0, 1, 0, -9139745, 10623846975]\) | \(66730743078481/60937500\) | \(77105377689600000000\) | \([2]\) | \(8257536\) | \(2.7397\) | |
| 162240.hj3 | 162240fc4 | \([0, 1, 0, -6111265, -5758412737]\) | \(19948814692561/231344100\) | \(292724089546840473600\) | \([2, 2]\) | \(8257536\) | \(2.7397\) | |
| 162240.hj4 | 162240fc5 | \([0, 1, 0, -1244065, -14674149697]\) | \(-168288035761/73415764890\) | \(-92894363572747897405440\) | \([2]\) | \(16515072\) | \(3.0863\) | |
| 162240.hj1 | 162240fc6 | \([0, 1, 0, -97506465, -370626330177]\) | \(81025909800741361/11088090\) | \(14029971155795312640\) | \([2]\) | \(16515072\) | \(3.0863\) |
Rank
sage: E.rank()
The elliptic curves in class 162240fc have rank \(1\).
Complex multiplication
The elliptic curves in class 162240fc do not have complex multiplication.Modular form 162240.2.a.fc
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
