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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 52983a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52983.e6 | 52983a1 | \([1, -1, 1, 7411, 52260]\) | \(103823/63\) | \(-27318450663567\) | \([2]\) | \(100352\) | \(1.2674\) | \(\Gamma_0(N)\)-optimal |
| 52983.e5 | 52983a2 | \([1, -1, 1, -30434, 445848]\) | \(7189057/3969\) | \(1721062391804721\) | \([2, 2]\) | \(200704\) | \(1.6140\) | |
| 52983.e3 | 52983a3 | \([1, -1, 1, -295349, -61332330]\) | \(6570725617/45927\) | \(19915150533740343\) | \([2]\) | \(401408\) | \(1.9606\) | |
| 52983.e2 | 52983a4 | \([1, -1, 1, -371039, 86959518]\) | \(13027640977/21609\) | \(9370228577603481\) | \([2, 2]\) | \(401408\) | \(1.9606\) | |
| 52983.e4 | 52983a5 | \([1, -1, 1, -257504, 141093006]\) | \(-4354703137/17294403\) | \(-7499306271608652627\) | \([2]\) | \(802816\) | \(2.3072\) | |
| 52983.e1 | 52983a6 | \([1, -1, 1, -5934254, 5565613650]\) | \(53297461115137/147\) | \(63743051548323\) | \([2]\) | \(802816\) | \(2.3072\) |
Rank
sage: E.rank()
The elliptic curves in class 52983a have rank \(0\).
Complex multiplication
The elliptic curves in class 52983a do not have complex multiplication.Modular form 52983.2.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.
