Show commands:
SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 67760.ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67760.ba1 | 67760ba4 | \([0, 0, 0, -518243, -143590942]\) | \(2121328796049/120050\) | \(871120478412800\) | \([2]\) | \(491520\) | \(1.9311\) | |
67760.ba2 | 67760ba3 | \([0, 0, 0, -169763, 25158562]\) | \(74565301329/5468750\) | \(39682966400000000\) | \([2]\) | \(491520\) | \(1.9311\) | |
67760.ba3 | 67760ba2 | \([0, 0, 0, -34243, -1972542]\) | \(611960049/122500\) | \(888898447360000\) | \([2, 2]\) | \(245760\) | \(1.5845\) | |
67760.ba4 | 67760ba1 | \([0, 0, 0, 4477, -183678]\) | \(1367631/2800\) | \(-20317678796800\) | \([2]\) | \(122880\) | \(1.2379\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 67760.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 67760.ba do not have complex multiplication.Modular form 67760.2.a.ba
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.