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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 32490.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32490.j1 | 32490j4 | \([1, -1, 0, -24294465, -44501105219]\) | \(46237740924063961/1806561830400\) | \(61958652518370484569600\) | \([2]\) | \(2488320\) | \(3.1406\) | |
32490.j2 | 32490j2 | \([1, -1, 0, -3582090, 2591768956]\) | \(148212258825961/1218375000\) | \(41785933917000375000\) | \([2]\) | \(829440\) | \(2.5913\) | |
32490.j3 | 32490j1 | \([1, -1, 0, -73170, 94119700]\) | \(-1263214441/110808000\) | \(-3800320726767192000\) | \([2]\) | \(414720\) | \(2.2447\) | \(\Gamma_0(N)\)-optimal |
32490.j4 | 32490j3 | \([1, -1, 0, 657855, -2526312515]\) | \(918046641959/80912056320\) | \(-2774996071386997063680\) | \([2]\) | \(1244160\) | \(2.7941\) |
Rank
sage: E.rank()
The elliptic curves in class 32490.j have rank \(1\).
Complex multiplication
The elliptic curves in class 32490.j do not have complex multiplication.Modular form 32490.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.