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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 408629.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
408629.a1 | 408629a1 | \([1, 0, 0, -110054, -14060285]\) | \(23320116793/2873\) | \(18161276039777\) | \([2]\) | \(1951488\) | \(1.5676\) | \(\Gamma_0(N)\)-optimal |
408629.a2 | 408629a2 | \([1, 0, 0, -100809, -16517606]\) | \(-17923019113/8254129\) | \(-52177346062279321\) | \([2]\) | \(3902976\) | \(1.9142\) |
Rank
sage: E.rank()
The elliptic curves in class 408629.a have rank \(1\).
Complex multiplication
The elliptic curves in class 408629.a do not have complex multiplication.Modular form 408629.2.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.