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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 33150.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33150.e1 | 33150a4 | \([1, 1, 0, -1473375, -688978125]\) | \(22638311752145721841/72499050\) | \(1132797656250\) | \([2]\) | \(294912\) | \(1.9617\) | |
| 33150.e2 | 33150a2 | \([1, 1, 0, -92125, -10784375]\) | \(5534056064805841/9890302500\) | \(154535976562500\) | \([2, 2]\) | \(147456\) | \(1.6151\) | |
| 33150.e3 | 33150a3 | \([1, 1, 0, -62875, -17716625]\) | \(-1759334717565361/7634341406250\) | \(-119286584472656250\) | \([2]\) | \(294912\) | \(1.9617\) | |
| 33150.e4 | 33150a1 | \([1, 1, 0, -7625, -52875]\) | \(3138428376721/1747933200\) | \(27311456250000\) | \([2]\) | \(73728\) | \(1.2685\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33150.e have rank \(1\).
Complex multiplication
The elliptic curves in class 33150.e do not have complex multiplication.Modular form 33150.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
