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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 63.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63.a1 | 63a5 | \([1, -1, 0, -7056, 229905]\) | \(53297461115137/147\) | \(107163\) | \([4]\) | \(32\) | \(0.62351\) | |
| 63.a2 | 63a4 | \([1, -1, 0, -441, 3672]\) | \(13027640977/21609\) | \(15752961\) | \([2, 2]\) | \(16\) | \(0.27694\) | |
| 63.a3 | 63a3 | \([1, -1, 0, -351, -2430]\) | \(6570725617/45927\) | \(33480783\) | \([2]\) | \(16\) | \(0.27694\) | |
| 63.a4 | 63a6 | \([1, -1, 0, -306, 5859]\) | \(-4354703137/17294403\) | \(-12607619787\) | \([2]\) | \(32\) | \(0.62351\) | |
| 63.a5 | 63a2 | \([1, -1, 0, -36, 27]\) | \(7189057/3969\) | \(2893401\) | \([2, 2]\) | \(8\) | \(-0.069636\) | |
| 63.a6 | 63a1 | \([1, -1, 0, 9, 0]\) | \(103823/63\) | \(-45927\) | \([2]\) | \(4\) | \(-0.41621\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 63.a have rank \(0\).
Complex multiplication
The elliptic curves in class 63.a do not have complex multiplication.Modular form 63.2.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 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.
