Show commands:
SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 15870.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15870.bg1 | 15870bg3 | \([1, 0, 0, -58401611, -171789912459]\) | \(148809678420065817601/20700\) | \(3064342902300\) | \([2]\) | \(811008\) | \(2.7208\) | |
| 15870.bg2 | 15870bg5 | \([1, 0, 0, -13664081, 16651803795]\) | \(1905890658841300321/293666194803750\) | \(43473136217020311783750\) | \([2]\) | \(1622016\) | \(3.0673\) | |
| 15870.bg3 | 15870bg4 | \([1, 0, 0, -3745331, -2537009955]\) | \(39248884582600321/3935264062500\) | \(582560313941939062500\) | \([2, 2]\) | \(811008\) | \(2.7208\) | |
| 15870.bg4 | 15870bg2 | \([1, 0, 0, -3650111, -2684429559]\) | \(36330796409313601/428490000\) | \(63431898077610000\) | \([2, 2]\) | \(405504\) | \(2.3742\) | |
| 15870.bg5 | 15870bg1 | \([1, 0, 0, -222191, -44245575]\) | \(-8194759433281/965779200\) | \(-142969982449708800\) | \([4]\) | \(202752\) | \(2.0276\) | \(\Gamma_0(N)\)-optimal |
| 15870.bg6 | 15870bg6 | \([1, 0, 0, 4649899, -12290588169]\) | \(75108181893694559/484313964843750\) | \(-71695848340759277343750\) | \([2]\) | \(1622016\) | \(3.0673\) |
Rank
sage: E.rank()
The elliptic curves in class 15870.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 15870.bg do not have complex multiplication.Modular form 15870.2.a.bg
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
