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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 147a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 147.a6 | 147a1 | \([1, 1, 1, 48, 48]\) | \(103823/63\) | \(-7411887\) | \([4]\) | \(24\) | \(0.0074397\) | \(\Gamma_0(N)\)-optimal |
| 147.a5 | 147a2 | \([1, 1, 1, -197, 146]\) | \(7189057/3969\) | \(466948881\) | \([2, 2]\) | \(48\) | \(0.35401\) | |
| 147.a3 | 147a3 | \([1, 1, 1, -1912, -32782]\) | \(6570725617/45927\) | \(5403265623\) | \([2]\) | \(96\) | \(0.70059\) | |
| 147.a2 | 147a4 | \([1, 1, 1, -2402, 44246]\) | \(13027640977/21609\) | \(2542277241\) | \([2, 2]\) | \(96\) | \(0.70059\) | |
| 147.a1 | 147a5 | \([1, 1, 1, -38417, 2882228]\) | \(53297461115137/147\) | \(17294403\) | \([2]\) | \(192\) | \(1.0472\) | |
| 147.a4 | 147a6 | \([1, 1, 1, -1667, 72764]\) | \(-4354703137/17294403\) | \(-2034669218547\) | \([2]\) | \(192\) | \(1.0472\) |
Rank
sage: E.rank()
The elliptic curves in class 147a have rank \(0\).
Complex multiplication
The elliptic curves in class 147a do not have complex multiplication.Modular form 147.2.a.a
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.
