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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 33600dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.fm6 | 33600dd1 | \([0, 1, 0, 1567, -4737]\) | \(103823/63\) | \(-258048000000\) | \([2]\) | \(32768\) | \(0.87892\) | \(\Gamma_0(N)\)-optimal |
| 33600.fm5 | 33600dd2 | \([0, 1, 0, -6433, -44737]\) | \(7189057/3969\) | \(16257024000000\) | \([2, 2]\) | \(65536\) | \(1.2255\) | |
| 33600.fm3 | 33600dd3 | \([0, 1, 0, -62433, 5947263]\) | \(6570725617/45927\) | \(188116992000000\) | \([2]\) | \(131072\) | \(1.5721\) | |
| 33600.fm2 | 33600dd4 | \([0, 1, 0, -78433, -8468737]\) | \(13027640977/21609\) | \(88510464000000\) | \([2, 2]\) | \(131072\) | \(1.5721\) | |
| 33600.fm4 | 33600dd5 | \([0, 1, 0, -54433, -13724737]\) | \(-4354703137/17294403\) | \(-70837874688000000\) | \([2]\) | \(262144\) | \(1.9186\) | |
| 33600.fm1 | 33600dd6 | \([0, 1, 0, -1254433, -541196737]\) | \(53297461115137/147\) | \(602112000000\) | \([2]\) | \(262144\) | \(1.9186\) |
Rank
sage: E.rank()
The elliptic curves in class 33600dd have rank \(1\).
Complex multiplication
The elliptic curves in class 33600dd do not have complex multiplication.Modular form 33600.2.a.dd
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.
