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SageMath
E = EllipticCurve("hn1")
E.isogeny_class()
Elliptic curves in class 88200.hn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88200.hn1 | 88200bb2 | \([0, 0, 0, -385875, -92181250]\) | \(1000188\) | \(6353046000000000\) | \([2]\) | \(737280\) | \(1.9517\) | |
88200.hn2 | 88200bb1 | \([0, 0, 0, -18375, -2143750]\) | \(-432\) | \(-1588261500000000\) | \([2]\) | \(368640\) | \(1.6052\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 88200.hn have rank \(1\).
Complex multiplication
The elliptic curves in class 88200.hn do not have complex multiplication.Modular form 88200.2.a.hn
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.