Show commands:
SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 82810.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82810.y1 | 82810w2 | \([1, -1, 0, -666899, -209456857]\) | \(-33669235266849/156250\) | \(-152226776406250\) | \([]\) | \(740880\) | \(1.9243\) | |
82810.y2 | 82810w1 | \([1, -1, 0, 1951, 18045]\) | \(842751/640\) | \(-623520876160\) | \([]\) | \(105840\) | \(0.95135\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 82810.y have rank \(0\).
Complex multiplication
The elliptic curves in class 82810.y do not have complex multiplication.Modular form 82810.2.a.y
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.