Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 36225bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36225.s1 | 36225bw1 | \([1, -1, 1, -36605, 2703772]\) | \(476196576129/197225\) | \(2246516015625\) | \([2]\) | \(110592\) | \(1.3320\) | \(\Gamma_0(N)\)-optimal |
36225.s2 | 36225bw2 | \([1, -1, 1, -30980, 3558772]\) | \(-288673724529/311181605\) | \(-3544552969453125\) | \([2]\) | \(221184\) | \(1.6786\) |
Rank
sage: E.rank()
The elliptic curves in class 36225bw have rank \(2\).
Complex multiplication
The elliptic curves in class 36225bw do not have complex multiplication.Modular form 36225.2.a.bw
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.