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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 55440.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.a1 | 55440j4 | \([0, 0, 0, -40683, 3157562]\) | \(4987755354962/1537305\) | \(2295184066560\) | \([2]\) | \(131072\) | \(1.3485\) | |
55440.a2 | 55440j2 | \([0, 0, 0, -2883, 35282]\) | \(3550014724/1334025\) | \(995844326400\) | \([2, 2]\) | \(65536\) | \(1.0019\) | |
55440.a3 | 55440j1 | \([0, 0, 0, -1263, -16882]\) | \(1193895376/31185\) | \(5819869440\) | \([2]\) | \(32768\) | \(0.65532\) | \(\Gamma_0(N)\)-optimal |
55440.a4 | 55440j3 | \([0, 0, 0, 8997, 251498]\) | \(53946017998/49520625\) | \(-73933896960000\) | \([2]\) | \(131072\) | \(1.3485\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.a have rank \(2\).
Complex multiplication
The elliptic curves in class 55440.a do not have complex multiplication.Modular form 55440.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.