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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 1666.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1666.m1 | 1666l4 | \([1, 1, 1, -5538, 107309]\) | \(159661140625/48275138\) | \(5679521710562\) | \([2]\) | \(3456\) | \(1.1524\) | |
1666.m2 | 1666l3 | \([1, 1, 1, -5048, 135925]\) | \(120920208625/19652\) | \(2312038148\) | \([2]\) | \(1728\) | \(0.80587\) | |
1666.m3 | 1666l2 | \([1, 1, 1, -2108, -38123]\) | \(8805624625/2312\) | \(272004488\) | \([2]\) | \(1152\) | \(0.60314\) | |
1666.m4 | 1666l1 | \([1, 1, 1, -148, -491]\) | \(3048625/1088\) | \(128002112\) | \([2]\) | \(576\) | \(0.25656\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1666.m have rank \(0\).
Complex multiplication
The elliptic curves in class 1666.m do not have complex multiplication.Modular form 1666.2.a.m
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.