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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 69696.fj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.fj1 | 69696gj2 | \([0, 0, 0, -53724, 2278672]\) | \(810448/363\) | \(7680866267086848\) | \([2]\) | \(368640\) | \(1.7435\) | |
69696.fj2 | 69696gj1 | \([0, 0, 0, 11616, 266200]\) | \(131072/99\) | \(-130923856825344\) | \([2]\) | \(184320\) | \(1.3970\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 69696.fj have rank \(0\).
Complex multiplication
The elliptic curves in class 69696.fj do not have complex multiplication.Modular form 69696.2.a.fj
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.