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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 333795ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333795.ca3 | 333795ca1 | \([1, 0, 1, -58818, 5481583]\) | \(932288503609/779625\) | \(18818252231625\) | \([2]\) | \(1327104\) | \(1.4751\) | \(\Gamma_0(N)\)-optimal |
333795.ca2 | 333795ca2 | \([1, 0, 1, -71823, 2875381]\) | \(1697509118089/833765625\) | \(20125075303265625\) | \([2, 2]\) | \(2654208\) | \(1.8216\) | |
333795.ca4 | 333795ca3 | \([1, 0, 1, 261972, 22101973]\) | \(82375335041831/56396484375\) | \(-1361274032958984375\) | \([2]\) | \(5308416\) | \(2.1682\) | |
333795.ca1 | 333795ca4 | \([1, 0, 1, -613698, -183096119]\) | \(1058993490188089/13182390375\) | \(318190857261498375\) | \([2]\) | \(5308416\) | \(2.1682\) |
Rank
sage: E.rank()
The elliptic curves in class 333795ca have rank \(0\).
Complex multiplication
The elliptic curves in class 333795ca do not have complex multiplication.Modular form 333795.2.a.ca
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 Cremona 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 Cremona labels.