Show commands:
SageMath
E = EllipticCurve("rw1")
E.isogeny_class()
Elliptic curves in class 436800.rw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
436800.rw1 | 436800rw3 | \([0, 1, 0, -170301633, -698762995137]\) | \(266716694084614489298/51372277695070605\) | \(105210424719504599040000000\) | \([2]\) | \(132120576\) | \(3.7103\) | \(\Gamma_0(N)\)-optimal* |
436800.rw2 | 436800rw2 | \([0, 1, 0, -161481633, -789847135137]\) | \(454771411897393003396/23468066028225\) | \(24031299612902400000000\) | \([2, 2]\) | \(66060288\) | \(3.3638\) | \(\Gamma_0(N)\)-optimal* |
436800.rw3 | 436800rw1 | \([0, 1, 0, -161479633, -789867677137]\) | \(1819018058610682173904/4844385\) | \(1240162560000000\) | \([2]\) | \(33030144\) | \(3.0172\) | \(\Gamma_0(N)\)-optimal* |
436800.rw4 | 436800rw4 | \([0, 1, 0, -152693633, -879616555137]\) | \(-192245661431796830258/51935513760073125\) | \(-106363932180629760000000000\) | \([2]\) | \(132120576\) | \(3.7103\) |
Rank
sage: E.rank()
The elliptic curves in class 436800.rw have rank \(0\).
Complex multiplication
The elliptic curves in class 436800.rw do not have complex multiplication.Modular form 436800.2.a.rw
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.