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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 19890.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19890.e1 | 19890e1 | \([1, -1, 0, -122580, -16299824]\) | \(279419703685750081/3666124800000\) | \(2672604979200000\) | \([2]\) | \(122880\) | \(1.7678\) | \(\Gamma_0(N)\)-optimal |
19890.e2 | 19890e2 | \([1, -1, 0, -18900, -43070000]\) | \(-1024222994222401/1098922500000000\) | \(-801114502500000000\) | \([2]\) | \(245760\) | \(2.1144\) |
Rank
sage: E.rank()
The elliptic curves in class 19890.e have rank \(1\).
Complex multiplication
The elliptic curves in class 19890.e do not have complex multiplication.Modular form 19890.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.