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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 8007.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8007.a1 | 8007a2 | \([1, 1, 0, -840, -9729]\) | \(65670324063625/3675213\) | \(3675213\) | \([2]\) | \(1920\) | \(0.32470\) | |
8007.a2 | 8007a1 | \([1, 1, 0, -55, -152]\) | \(18927429625/3771297\) | \(3771297\) | \([2]\) | \(960\) | \(-0.021869\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8007.a have rank \(1\).
Complex multiplication
The elliptic curves in class 8007.a do not have complex multiplication.Modular form 8007.2.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.