Show commands:
SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 388080.dj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.dj1 | 388080dj4 | \([0, 0, 0, -184274643, 935540391058]\) | \(1969902499564819009/63690429687500\) | \(22374322581996000000000000\) | \([2]\) | \(95551488\) | \(3.6381\) | |
| 388080.dj2 | 388080dj2 | \([0, 0, 0, -25232403, -48351607598]\) | \(5057359576472449/51765560000\) | \(18185139333499944960000\) | \([2]\) | \(31850496\) | \(3.0888\) | |
| 388080.dj3 | 388080dj1 | \([0, 0, 0, -395283, -1861486382]\) | \(-19443408769/4249907200\) | \(-1492984033910666035200\) | \([2]\) | \(15925248\) | \(2.7422\) | \(\Gamma_0(N)\)-optimal |
| 388080.dj4 | 388080dj3 | \([0, 0, 0, 3556077, 50143943122]\) | \(14156681599871/3100231750000\) | \(-1089105311328427008000000\) | \([2]\) | \(47775744\) | \(3.2915\) |
Rank
sage: E.rank()
The elliptic curves in class 388080.dj have rank \(0\).
Complex multiplication
The elliptic curves in class 388080.dj do not have complex multiplication.Modular form 388080.2.a.dj
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.
