Show commands:
SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 278850.hw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
278850.hw1 | 278850hw2 | \([1, 0, 0, -123902438, -530855410008]\) | \(2789222297765780449/677605500\) | \(51104255091398437500\) | \([2]\) | \(37158912\) | \(3.1590\) | |
278850.hw2 | 278850hw1 | \([1, 0, 0, -7714938, -8360222508]\) | \(-673350049820449/10617750000\) | \(-800778925933593750000\) | \([2]\) | \(18579456\) | \(2.8124\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 278850.hw have rank \(0\).
Complex multiplication
The elliptic curves in class 278850.hw do not have complex multiplication.Modular form 278850.2.a.hw
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.