Show commands:
SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 4032bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.n3 | 4032bk1 | \([0, 0, 0, -111, 376]\) | \(3241792/567\) | \(26453952\) | \([2]\) | \(1024\) | \(0.14410\) | \(\Gamma_0(N)\)-optimal |
| 4032.n2 | 4032bk2 | \([0, 0, 0, -516, -4160]\) | \(5088448/441\) | \(1316818944\) | \([2, 2]\) | \(2048\) | \(0.49067\) | |
| 4032.n1 | 4032bk3 | \([0, 0, 0, -8076, -279344]\) | \(2438569736/21\) | \(501645312\) | \([2]\) | \(4096\) | \(0.83725\) | |
| 4032.n4 | 4032bk4 | \([0, 0, 0, 564, -19280]\) | \(830584/7203\) | \(-172064342016\) | \([4]\) | \(4096\) | \(0.83725\) |
Rank
sage: E.rank()
The elliptic curves in class 4032bk have rank \(0\).
Complex multiplication
The elliptic curves in class 4032bk do not have complex multiplication.Modular form 4032.2.a.bk
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.
