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SageMath
sage: E = EllipticCurve("f1")
sage: E.isogeny_class()
Elliptic curves in class 441.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 441.f1 | 441c5 | \([1, -1, 0, -345753, -78165914]\) | \(53297461115137/147\) | \(12607619787\) | \([2]\) | \(1536\) | \(1.5965\) | |
| 441.f2 | 441c3 | \([1, -1, 0, -21618, -1216265]\) | \(13027640977/21609\) | \(1853320108689\) | \([2, 2]\) | \(768\) | \(1.2499\) | |
| 441.f3 | 441c4 | \([1, -1, 0, -17208, 867901]\) | \(6570725617/45927\) | \(3938980639167\) | \([2]\) | \(768\) | \(1.2499\) | |
| 441.f4 | 441c6 | \([1, -1, 0, -15003, -1979636]\) | \(-4354703137/17294403\) | \(-1483273860320763\) | \([2]\) | \(1536\) | \(1.5965\) | |
| 441.f5 | 441c2 | \([1, -1, 0, -1773, -5720]\) | \(7189057/3969\) | \(340405734249\) | \([2, 2]\) | \(384\) | \(0.90332\) | |
| 441.f6 | 441c1 | \([1, -1, 0, 432, -869]\) | \(103823/63\) | \(-5403265623\) | \([2]\) | \(192\) | \(0.55675\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 441.f have rank \(1\).
Complex multiplication
The elliptic curves in class 441.f do not have complex multiplication.Modular form 441.2.a.f
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
