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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 3600.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3600.f1 | 3600bk7 | \([0, 0, 0, -19200675, -32383430750]\) | \(16778985534208729/81000\) | \(3779136000000000\) | \([2]\) | \(110592\) | \(2.6113\) | |
3600.f2 | 3600bk8 | \([0, 0, 0, -1632675, -109286750]\) | \(10316097499609/5859375000\) | \(273375000000000000000\) | \([2]\) | \(110592\) | \(2.6113\) | |
3600.f3 | 3600bk6 | \([0, 0, 0, -1200675, -505430750]\) | \(4102915888729/9000000\) | \(419904000000000000\) | \([2, 2]\) | \(55296\) | \(2.2648\) | |
3600.f4 | 3600bk5 | \([0, 0, 0, -1038675, 407439250]\) | \(2656166199049/33750\) | \(1574640000000000\) | \([2]\) | \(36864\) | \(2.0620\) | |
3600.f5 | 3600bk4 | \([0, 0, 0, -246675, -40616750]\) | \(35578826569/5314410\) | \(247949112960000000\) | \([2]\) | \(36864\) | \(2.0620\) | |
3600.f6 | 3600bk2 | \([0, 0, 0, -66675, 6003250]\) | \(702595369/72900\) | \(3401222400000000\) | \([2, 2]\) | \(18432\) | \(1.7155\) | |
3600.f7 | 3600bk3 | \([0, 0, 0, -48675, -13526750]\) | \(-273359449/1536000\) | \(-71663616000000000\) | \([2]\) | \(27648\) | \(1.9182\) | |
3600.f8 | 3600bk1 | \([0, 0, 0, 5325, 459250]\) | \(357911/2160\) | \(-100776960000000\) | \([2]\) | \(9216\) | \(1.3689\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3600.f have rank \(1\).
Complex multiplication
The elliptic curves in class 3600.f do not have complex multiplication.Modular form 3600.2.a.f
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.