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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 66.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66.a1 | 66a3 | \([1, 0, 1, -81, -284]\) | \(57736239625/255552\) | \(255552\) | \([2]\) | \(12\) | \(-0.11008\) | |
66.a2 | 66a4 | \([1, 0, 1, -41, -556]\) | \(-7357983625/127552392\) | \(-127552392\) | \([2]\) | \(24\) | \(0.23650\) | |
66.a3 | 66a1 | \([1, 0, 1, -6, 4]\) | \(18609625/1188\) | \(1188\) | \([6]\) | \(4\) | \(-0.65938\) | \(\Gamma_0(N)\)-optimal |
66.a4 | 66a2 | \([1, 0, 1, 4, 20]\) | \(9938375/176418\) | \(-176418\) | \([6]\) | \(8\) | \(-0.31281\) |
Rank
sage: E.rank()
The elliptic curves in class 66.a have rank \(0\).
Complex multiplication
The elliptic curves in class 66.a do not have complex multiplication.Modular form 66.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.