Show commands:
SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 5040.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.bm1 | 5040bn3 | \([0, 0, 0, -38547, -2912814]\) | \(2121328796049/120050\) | \(358467379200\) | \([2]\) | \(12288\) | \(1.2814\) | |
5040.bm2 | 5040bn4 | \([0, 0, 0, -12627, 510354]\) | \(74565301329/5468750\) | \(16329600000000\) | \([2]\) | \(12288\) | \(1.2814\) | |
5040.bm3 | 5040bn2 | \([0, 0, 0, -2547, -40014]\) | \(611960049/122500\) | \(365783040000\) | \([2, 2]\) | \(6144\) | \(0.93487\) | |
5040.bm4 | 5040bn1 | \([0, 0, 0, 333, -3726]\) | \(1367631/2800\) | \(-8360755200\) | \([2]\) | \(3072\) | \(0.58829\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5040.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.bm do not have complex multiplication.Modular form 5040.2.a.bm
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.