Show commands:
SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 489762.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.bm1 | 489762bm4 | \([1, -1, 0, -2306582082, 42432281377812]\) | \(385693937170561837203625/2159357734550274048\) | \(7598226556215870947240214528\) | \([2]\) | \(530841600\) | \(4.1921\) | \(\Gamma_0(N)\)-optimal* |
489762.bm2 | 489762bm2 | \([1, -1, 0, -170345187, -815284208907]\) | \(155355156733986861625/8291568305839392\) | \(29175904245077700468033312\) | \([2]\) | \(176947200\) | \(3.6428\) | \(\Gamma_0(N)\)-optimal* |
489762.bm3 | 489762bm3 | \([1, -1, 0, -63776322, 1398355754004]\) | \(-8152944444844179625/235342826399858688\) | \(-828111102090608849681645568\) | \([2]\) | \(265420800\) | \(3.8456\) | \(\Gamma_0(N)\)-optimal* |
489762.bm4 | 489762bm1 | \([1, -1, 0, 7064253, -50897895723]\) | \(11079872671250375/324440155855872\) | \(-1141621774235717215214592\) | \([2]\) | \(88473600\) | \(3.2963\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 489762.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 489762.bm do not have complex multiplication.Modular form 489762.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.