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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 158400fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 158400.ku4 | 158400fc1 | \([0, 0, 0, 196800, -123469000]\) | \(72268906496/606436875\) | \(-7073479710000000000\) | \([2]\) | \(1769472\) | \(2.2983\) | \(\Gamma_0(N)\)-optimal |
| 158400.ku3 | 158400fc2 | \([0, 0, 0, -2840700, -1696894000]\) | \(13584145739344/1195803675\) | \(223165665043200000000\) | \([2]\) | \(3538944\) | \(2.6448\) | |
| 158400.ku2 | 158400fc3 | \([0, 0, 0, -14059200, -20306401000]\) | \(-26348629355659264/24169921875\) | \(-281917968750000000000\) | \([2]\) | \(5308416\) | \(2.8476\) | |
| 158400.ku1 | 158400fc4 | \([0, 0, 0, -224996700, -1299009526000]\) | \(6749703004355978704/5671875\) | \(1058508000000000000\) | \([2]\) | \(10616832\) | \(3.1942\) |
Rank
sage: E.rank()
The elliptic curves in class 158400fc have rank \(1\).
Complex multiplication
The elliptic curves in class 158400fc do not have complex multiplication.Modular form 158400.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
