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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 16650k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16650.z2 | 16650k1 | \([1, -1, 0, 28458, 5036116]\) | \(223759095911/1094104800\) | \(-12462537487500000\) | \([]\) | \(138240\) | \(1.7707\) | \(\Gamma_0(N)\)-optimal |
16650.z1 | 16650k2 | \([1, -1, 0, -1594917, 776638741]\) | \(-39390416456458249/56832000000\) | \(-647352000000000000\) | \([]\) | \(414720\) | \(2.3200\) |
Rank
sage: E.rank()
The elliptic curves in class 16650k have rank \(0\).
Complex multiplication
The elliptic curves in class 16650k do not have complex multiplication.Modular form 16650.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.