Show commands:
SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 72450.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.bd1 | 72450bw2 | \([1, -1, 0, -14022, 639036]\) | \(3346058125493/21595896\) | \(1967926023000\) | \([2]\) | \(184320\) | \(1.1954\) | |
72450.bd2 | 72450bw1 | \([1, -1, 0, -1422, -3564]\) | \(3491055413/1947456\) | \(177461928000\) | \([2]\) | \(92160\) | \(0.84880\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.bd do not have complex multiplication.Modular form 72450.2.a.bd
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.