Show commands:
SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 50400.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50400.df1 | 50400ck2 | \([0, 0, 0, -2700, 44000]\) | \(1259712/245\) | \(423360000000\) | \([2]\) | \(49152\) | \(0.94785\) | |
50400.df2 | 50400ck1 | \([0, 0, 0, -825, -8500]\) | \(2299968/175\) | \(4725000000\) | \([2]\) | \(24576\) | \(0.60127\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 50400.df have rank \(1\).
Complex multiplication
The elliptic curves in class 50400.df do not have complex multiplication.Modular form 50400.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.