Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 388416.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388416.bf1 | 388416bf3 | \([0, -1, 0, -93941569, 350488256833]\) | \(14489843500598257/6246072\) | \(39522138555408187392\) | \([2]\) | \(42467328\) | \(3.1029\) | |
388416.bf2 | 388416bf4 | \([0, -1, 0, -12559169, -9051196095]\) | \(34623662831857/14438442312\) | \(91359516441554328748032\) | \([2]\) | \(42467328\) | \(3.1029\) | |
388416.bf3 | 388416bf2 | \([0, -1, 0, -5900609, 5420518209]\) | \(3590714269297/73410624\) | \(464507109935167832064\) | \([2, 2]\) | \(21233664\) | \(2.7563\) | |
388416.bf4 | 388416bf1 | \([0, -1, 0, 18111, 253475649]\) | \(103823/4386816\) | \(-27757661097899851776\) | \([2]\) | \(10616832\) | \(2.4097\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388416.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 388416.bf do not have complex multiplication.Modular form 388416.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.