Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 479370cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
479370.cm2 | 479370cm1 | \([1, 0, 0, -66036, -22915440]\) | \(-53540005609/350208000\) | \(-208311885600768000\) | \([2]\) | \(8128512\) | \(2.0066\) | \(\Gamma_0(N)\)-optimal* |
479370.cm1 | 479370cm2 | \([1, 0, 0, -1680756, -837057264]\) | \(882774443450089/2166000000\) | \(1288387313286000000\) | \([2]\) | \(16257024\) | \(2.3532\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 479370cm have rank \(1\).
Complex multiplication
The elliptic curves in class 479370cm do not have complex multiplication.Modular form 479370.2.a.cm
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.