Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 323400bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
323400.bf3 | 323400bf1 | \([0, -1, 0, -289508, -28940988]\) | \(5702413264/2525985\) | \(1188718437060000000\) | \([2]\) | \(4718592\) | \(2.1640\) | \(\Gamma_0(N)\)-optimal |
323400.bf2 | 323400bf2 | \([0, -1, 0, -2274008, 1300674012]\) | \(690862540036/12006225\) | \(22600325840400000000\) | \([2, 2]\) | \(9437184\) | \(2.5105\) | |
323400.bf1 | 323400bf3 | \([0, -1, 0, -36231008, 83952012012]\) | \(1397097631688978/433125\) | \(1630615140000000000\) | \([2]\) | \(18874368\) | \(2.8571\) | |
323400.bf4 | 323400bf4 | \([0, -1, 0, -69008, 3712944012]\) | \(-9653618/1581886845\) | \(-5955436973676960000000\) | \([2]\) | \(18874368\) | \(2.8571\) |
Rank
sage: E.rank()
The elliptic curves in class 323400bf have rank \(1\).
Complex multiplication
The elliptic curves in class 323400bf do not have complex multiplication.Modular form 323400.2.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 Cremona 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 Cremona labels.