Show commands:
SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 84150.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.df1 | 84150bk8 | \([1, -1, 0, -45278208567, 3708371995205341]\) | \(901247067798311192691198986281/552431869440\) | \(6292544262840000000\) | \([2]\) | \(127401984\) | \(4.3148\) | |
| 84150.df2 | 84150bk7 | \([1, -1, 0, -2848896567, 57125962949341]\) | \(224494757451893010998773801/6152490825146276160000\) | \(70080715805181801885000000000\) | \([2]\) | \(127401984\) | \(4.3148\) | |
| 84150.df3 | 84150bk6 | \([1, -1, 0, -2829888567, 57943820165341]\) | \(220031146443748723000125481/172266701724057600\) | \(1962225399325593600000000\) | \([2, 2]\) | \(63700992\) | \(3.9682\) | |
| 84150.df4 | 84150bk5 | \([1, -1, 0, -559104192, 5084876362216]\) | \(1696892787277117093383481/1440538624914939000\) | \(16408635274421727046875000\) | \([2]\) | \(42467328\) | \(3.7655\) | |
| 84150.df5 | 84150bk4 | \([1, -1, 0, -366162192, -2667956503784]\) | \(476646772170172569823801/5862293314453125000\) | \(66775184784942626953125000\) | \([2]\) | \(42467328\) | \(3.7655\) | |
| 84150.df6 | 84150bk3 | \([1, -1, 0, -175680567, 918161285341]\) | \(-52643812360427830814761/1504091705903677440\) | \(-17132544587559075840000000\) | \([2]\) | \(31850496\) | \(3.6217\) | |
| 84150.df7 | 84150bk2 | \([1, -1, 0, -42729192, 41441737216]\) | \(757443433548897303481/373234243041000000\) | \(4251371299638890625000000\) | \([2, 2]\) | \(21233664\) | \(3.4189\) | |
| 84150.df8 | 84150bk1 | \([1, -1, 0, 9758808, 4962577216]\) | \(9023321954633914439/6156756739584000\) | \(-70129307236824000000000\) | \([2]\) | \(10616832\) | \(3.0724\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84150.df have rank \(0\).
Complex multiplication
The elliptic curves in class 84150.df do not have complex multiplication.Modular form 84150.2.a.df
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
