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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 150150gn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 150150.e7 | 150150gn1 | \([1, 1, 0, -18736525, 32987180125]\) | \(-46555485820017544148689/3157693080314572800\) | \(-49338954379915200000000\) | \([2]\) | \(18579456\) | \(3.1059\) | \(\Gamma_0(N)\)-optimal |
| 150150.e6 | 150150gn2 | \([1, 1, 0, -304504525, 2045079668125]\) | \(199841159336796255944706769/834505270358760000\) | \(13039144849355625000000\) | \([2, 2]\) | \(37158912\) | \(3.4525\) | |
| 150150.e8 | 150150gn3 | \([1, 1, 0, 105517475, 36973838125]\) | \(8315279469612171276463151/4849789796887785750000\) | \(-75777965576371652343750000\) | \([2]\) | \(55738368\) | \(3.6552\) | |
| 150150.e2 | 150150gn4 | \([1, 1, 0, -4872067525, 130891464335125]\) | \(818546927584539194367471866449/14273634375000\) | \(223025537109375000\) | \([2]\) | \(74317824\) | \(3.7991\) | |
| 150150.e5 | 150150gn5 | \([1, 1, 0, -309229525, 1978329593125]\) | \(209289070072300727183442769/12893854589717635333800\) | \(201466477964338052090625000\) | \([2]\) | \(74317824\) | \(3.7991\) | |
| 150150.e4 | 150150gn6 | \([1, 1, 0, -423903025, 295860462625]\) | \(539142086340577084766074129/309580507925165039062500\) | \(4837195436330703735351562500\) | \([2, 2]\) | \(111476736\) | \(4.0018\) | |
| 150150.e1 | 150150gn7 | \([1, 1, 0, -4875099775, 130720376434375]\) | \(820076206880893214178646273009/2122496008872985839843750\) | \(33164000138640403747558593750\) | \([2]\) | \(222953472\) | \(4.3484\) | |
| 150150.e3 | 150150gn8 | \([1, 1, 0, -4443434275, -113549323131125]\) | \(620954771108295351491118574129/2882378618771462717156250\) | \(45037165918304104955566406250\) | \([2]\) | \(222953472\) | \(4.3484\) |
Rank
sage: E.rank()
The elliptic curves in class 150150gn have rank \(1\).
Complex multiplication
The elliptic curves in class 150150gn do not have complex multiplication.Modular form 150150.2.a.gn
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
