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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 330.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
330.a1 | 330a3 | \([1, 1, 0, -4163, 77343]\) | \(7981893677157049/1917731420550\) | \(1917731420550\) | \([2]\) | \(640\) | \(1.0680\) | |
330.a2 | 330a2 | \([1, 1, 0, -1413, -20007]\) | \(312341975961049/17862322500\) | \(17862322500\) | \([2, 2]\) | \(320\) | \(0.72141\) | |
330.a3 | 330a1 | \([1, 1, 0, -1393, -20603]\) | \(299270638153369/1069200\) | \(1069200\) | \([2]\) | \(160\) | \(0.37484\) | \(\Gamma_0(N)\)-optimal |
330.a4 | 330a4 | \([1, 1, 0, 1017, -78813]\) | \(116149984977671/2779502343750\) | \(-2779502343750\) | \([2]\) | \(640\) | \(1.0680\) |
Rank
sage: E.rank()
The elliptic curves in class 330.a have rank \(0\).
Complex multiplication
The elliptic curves in class 330.a do not have complex multiplication.Modular form 330.2.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.