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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 429.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
429.a1 | 429a2 | \([1, 1, 1, -13, 8]\) | \(244140625/61347\) | \(61347\) | \([2]\) | \(32\) | \(-0.37136\) | |
429.a2 | 429a1 | \([1, 1, 1, 2, 2]\) | \(857375/1287\) | \(-1287\) | \([2]\) | \(16\) | \(-0.71794\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 429.a have rank \(1\).
Complex multiplication
The elliptic curves in class 429.a do not have complex multiplication.Modular form 429.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.