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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 333795.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333795.bm1 | 333795bm3 | \([1, 1, 0, -166180063, 824479800718]\) | \(21026497979043461623321/161783881875\) | \(3905069611845661875\) | \([2]\) | \(39321600\) | \(3.1599\) | |
333795.bm2 | 333795bm2 | \([1, 1, 0, -10393168, 12861235147]\) | \(5143681768032498601/14238434358225\) | \(343681191773626655025\) | \([2, 2]\) | \(19660800\) | \(2.8133\) | |
333795.bm3 | 333795bm4 | \([1, 1, 0, -6296593, 23108407852]\) | \(-1143792273008057401/8897444448004035\) | \(-214762679287364307090915\) | \([2]\) | \(39321600\) | \(3.1599\) | |
333795.bm4 | 333795bm1 | \([1, 1, 0, -912523, 22545688]\) | \(3481467828171481/2005331497785\) | \(48403827395658784665\) | \([2]\) | \(9830400\) | \(2.4667\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333795.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 333795.bm do not have complex multiplication.Modular form 333795.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.