Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 485520.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.a1 | 485520a3 | \([0, -1, 0, -4457073696, 114532428600720]\) | \(396168254899399897286404/722925\) | \(17868443718988800\) | \([4]\) | \(141557760\) | \(3.7484\) | \(\Gamma_0(N)\)-optimal* |
485520.a2 | 485520a2 | \([0, -1, 0, -278567196, 1789637620320]\) | \(386883437712133521616/522620555625\) | \(3229386168887494560000\) | \([2, 2]\) | \(70778880\) | \(3.4018\) | \(\Gamma_0(N)\)-optimal* |
485520.a3 | 485520a4 | \([0, -1, 0, -276110696, 1822747309920]\) | \(-94184605035375674404/3558001972390425\) | \(-87942674545367017857868800\) | \([2]\) | \(141557760\) | \(3.7484\) | |
485520.a4 | 485520a1 | \([0, -1, 0, -17564071, 27448921570]\) | \(1551621461335545856/55495789453125\) | \(21432535154148431250000\) | \([2]\) | \(35389440\) | \(3.0553\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485520.a have rank \(1\).
Complex multiplication
The elliptic curves in class 485520.a do not have complex multiplication.Modular form 485520.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.