Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 63525.bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
63525.bw1 | 63525q4 | \([1, 1, 0, -340375, 76286500]\) | \(157551496201/13125\) | \(363308408203125\) | \([2]\) | \(552960\) | \(1.8377\) | |
63525.bw2 | 63525q2 | \([1, 1, 0, -22750, 1009375]\) | \(47045881/11025\) | \(305179062890625\) | \([2, 2]\) | \(276480\) | \(1.4911\) | |
63525.bw3 | 63525q1 | \([1, 1, 0, -7625, -246000]\) | \(1771561/105\) | \(2906467265625\) | \([2]\) | \(138240\) | \(1.1446\) | \(\Gamma_0(N)\)-optimal |
63525.bw4 | 63525q3 | \([1, 1, 0, 52875, 6378750]\) | \(590589719/972405\) | \(-26916793346953125\) | \([2]\) | \(552960\) | \(1.8377\) |
Rank
sage: E.rank()
The elliptic curves in class 63525.bw have rank \(1\).
Complex multiplication
The elliptic curves in class 63525.bw do not have complex multiplication.Modular form 63525.2.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 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.