Show commands:
SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 30912bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30912.s2 | 30912bm1 | \([0, -1, 0, -6273, 359073]\) | \(-416618810500/598934007\) | \(-39251739082752\) | \([2]\) | \(86016\) | \(1.3005\) | \(\Gamma_0(N)\)-optimal |
30912.s1 | 30912bm2 | \([0, -1, 0, -122913, 16618689]\) | \(1566789944863250/925924041\) | \(121362715901952\) | \([2]\) | \(172032\) | \(1.6471\) |
Rank
sage: E.rank()
The elliptic curves in class 30912bm have rank \(1\).
Complex multiplication
The elliptic curves in class 30912bm do not have complex multiplication.Modular form 30912.2.a.bm
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.