Show commands:
SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 250470.em
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
250470.em1 | 250470em2 | \([1, -1, 1, -855977, -210897871]\) | \(53706380371489/16171875000\) | \(20885458561171875000\) | \([2]\) | \(5376000\) | \(2.4124\) | |
250470.em2 | 250470em1 | \([1, -1, 1, 145903, -22143679]\) | \(265971760991/317400000\) | \(-409911933360600000\) | \([2]\) | \(2688000\) | \(2.0658\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 250470.em have rank \(1\).
Complex multiplication
The elliptic curves in class 250470.em do not have complex multiplication.Modular form 250470.2.a.em
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.