Show commands:
SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 86640bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86640.n2 | 86640bz1 | \([0, -1, 0, -1220661, 519488640]\) | \(267219216891904/3655125\) | \(2751337212642000\) | \([2]\) | \(829440\) | \(2.1044\) | \(\Gamma_0(N)\)-optimal |
86640.n1 | 86640bz2 | \([0, -1, 0, -1254956, 488787756]\) | \(18148802937424/1947796875\) | \(23458769918316000000\) | \([2]\) | \(1658880\) | \(2.4510\) |
Rank
sage: E.rank()
The elliptic curves in class 86640bz have rank \(1\).
Complex multiplication
The elliptic curves in class 86640bz do not have complex multiplication.Modular form 86640.2.a.bz
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.