Show commands:
SageMath
E = EllipticCurve("fz1")
E.isogeny_class()
Elliptic curves in class 436800.fz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
436800.fz1 | 436800fz4 | \([0, -1, 0, -667233, -209553663]\) | \(8020417344913/187278\) | \(767090688000000\) | \([2]\) | \(4718592\) | \(1.9684\) | |
436800.fz2 | 436800fz2 | \([0, -1, 0, -43233, -3009663]\) | \(2181825073/298116\) | \(1221083136000000\) | \([2, 2]\) | \(2359296\) | \(1.6218\) | |
436800.fz3 | 436800fz1 | \([0, -1, 0, -11233, 414337]\) | \(38272753/4368\) | \(17891328000000\) | \([2]\) | \(1179648\) | \(1.2753\) | \(\Gamma_0(N)\)-optimal* |
436800.fz4 | 436800fz3 | \([0, -1, 0, 68767, -16113663]\) | \(8780064047/32388174\) | \(-132661960704000000\) | \([2]\) | \(4718592\) | \(1.9684\) |
Rank
sage: E.rank()
The elliptic curves in class 436800.fz have rank \(1\).
Complex multiplication
The elliptic curves in class 436800.fz do not have complex multiplication.Modular form 436800.2.a.fz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.