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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 142.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
142.e1 | 142d2 | \([1, 0, 0, -58, -170]\) | \(21601086625/715822\) | \(715822\) | \([]\) | \(12\) | \(-0.10354\) | |
142.e2 | 142d1 | \([1, 0, 0, -8, 8]\) | \(57066625/568\) | \(568\) | \([3]\) | \(4\) | \(-0.65284\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 142.e have rank \(0\).
Complex multiplication
The elliptic curves in class 142.e do not have complex multiplication.Modular form 142.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.