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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 6600.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6600.a1 | 6600v4 | \([0, -1, 0, -17608, 905212]\) | \(37736227588/33\) | \(528000000\) | \([2]\) | \(12288\) | \(0.97404\) | |
6600.a2 | 6600v3 | \([0, -1, 0, -2608, -30788]\) | \(122657188/43923\) | \(702768000000\) | \([2]\) | \(12288\) | \(0.97404\) | |
6600.a3 | 6600v2 | \([0, -1, 0, -1108, 14212]\) | \(37642192/1089\) | \(4356000000\) | \([2, 2]\) | \(6144\) | \(0.62747\) | |
6600.a4 | 6600v1 | \([0, -1, 0, 17, 712]\) | \(2048/891\) | \(-222750000\) | \([2]\) | \(3072\) | \(0.28089\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6600.a have rank \(2\).
Complex multiplication
The elliptic curves in class 6600.a do not have complex multiplication.Modular form 6600.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.