Show commands:
SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 212160.bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212160.bj1 | 212160hr2 | \([0, -1, 0, -6321, 188721]\) | \(1705021456336/68471325\) | \(1121834188800\) | \([2]\) | \(393216\) | \(1.0784\) | |
212160.bj2 | 212160hr1 | \([0, -1, 0, 179, 10621]\) | \(615962624/48481875\) | \(-49645440000\) | \([2]\) | \(196608\) | \(0.73186\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 212160.bj have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.bj do not have complex multiplication.Modular form 212160.2.a.bj
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.