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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 127050.cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127050.cr1 | 127050cv4 | \([1, 0, 1, -1129901, -462377302]\) | \(5763259856089/5670\) | \(156949232343750\) | \([2]\) | \(1966080\) | \(2.0173\) | |
127050.cr2 | 127050cv2 | \([1, 0, 1, -71151, -7114802]\) | \(1439069689/44100\) | \(1220716251562500\) | \([2, 2]\) | \(983040\) | \(1.6707\) | |
127050.cr3 | 127050cv1 | \([1, 0, 1, -10651, 266198]\) | \(4826809/1680\) | \(46503476250000\) | \([2]\) | \(491520\) | \(1.3241\) | \(\Gamma_0(N)\)-optimal |
127050.cr4 | 127050cv3 | \([1, 0, 1, 19599, -23994302]\) | \(30080231/9003750\) | \(-249229568027343750\) | \([2]\) | \(1966080\) | \(2.0173\) |
Rank
sage: E.rank()
The elliptic curves in class 127050.cr have rank \(1\).
Complex multiplication
The elliptic curves in class 127050.cr do not have complex multiplication.Modular form 127050.2.a.cr
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.