Show commands:
SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 8112.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8112.bg1 | 8112m3 | \([0, 1, 0, -51432, 4362372]\) | \(3044193988/85293\) | \(421573652517888\) | \([4]\) | \(43008\) | \(1.5853\) | |
| 8112.bg2 | 8112m2 | \([0, 1, 0, -7492, -154660]\) | \(37642192/13689\) | \(16914992230656\) | \([2, 2]\) | \(21504\) | \(1.2387\) | |
| 8112.bg3 | 8112m1 | \([0, 1, 0, -6647, -210768]\) | \(420616192/117\) | \(9035786448\) | \([2]\) | \(10752\) | \(0.89216\) | \(\Gamma_0(N)\)-optimal |
| 8112.bg4 | 8112m4 | \([0, 1, 0, 22928, -1067260]\) | \(269676572/257049\) | \(-1270503860880384\) | \([2]\) | \(43008\) | \(1.5853\) |
Rank
sage: E.rank()
The elliptic curves in class 8112.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 8112.bg do not have complex multiplication.Modular form 8112.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
