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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 33810.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33810.cd1 | 33810ca4 | \([1, 1, 1, -5409601, -4845047677]\) | \(148809678420065817601/20700\) | \(2435334300\) | \([2]\) | \(589824\) | \(2.1260\) | |
| 33810.cd2 | 33810ca6 | \([1, 1, 1, -1265671, 468935879]\) | \(1905890658841300321/293666194803750\) | \(34549534152466383750\) | \([2]\) | \(1179648\) | \(2.4725\) | |
| 33810.cd3 | 33810ca3 | \([1, 1, 1, -346921, -71656621]\) | \(39248884582600321/3935264062500\) | \(462979881689062500\) | \([2, 2]\) | \(589824\) | \(2.1260\) | |
| 33810.cd4 | 33810ca2 | \([1, 1, 1, -338101, -75809077]\) | \(36330796409313601/428490000\) | \(50411420010000\) | \([2, 2]\) | \(294912\) | \(1.7794\) | |
| 33810.cd5 | 33810ca1 | \([1, 1, 1, -20581, -1255381]\) | \(-8194759433281/965779200\) | \(-113622957100800\) | \([2]\) | \(147456\) | \(1.4328\) | \(\Gamma_0(N)\)-optimal |
| 33810.cd6 | 33810ca5 | \([1, 1, 1, 430709, -346315537]\) | \(75108181893694559/484313964843750\) | \(-56979053649902343750\) | \([2]\) | \(1179648\) | \(2.4725\) |
Rank
sage: E.rank()
The elliptic curves in class 33810.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 33810.cd do not have complex multiplication.Modular form 33810.2.a.cd
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
