Show commands:
SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 137904bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137904.r2 | 137904bk1 | \([0, -1, 0, -89288, 9766704]\) | \(3981876625/232713\) | \(4600877886738432\) | \([2]\) | \(688128\) | \(1.7589\) | \(\Gamma_0(N)\)-optimal |
137904.r1 | 137904bk2 | \([0, -1, 0, -265048, -40289744]\) | \(104154702625/24649677\) | \(487339142310678528\) | \([2]\) | \(1376256\) | \(2.1055\) |
Rank
sage: E.rank()
The elliptic curves in class 137904bk have rank \(1\).
Complex multiplication
The elliptic curves in class 137904bk do not have complex multiplication.Modular form 137904.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.