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SageMath
E = EllipticCurve("hq1")
E.isogeny_class()
Elliptic curves in class 338800.hq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
338800.hq1 | 338800hq2 | \([0, -1, 0, -297458, -62414713]\) | \(-262885120/343\) | \(-3797783893750000\) | \([]\) | \(2916000\) | \(1.8965\) | |
338800.hq2 | 338800hq1 | \([0, -1, 0, 5042, -402213]\) | \(1280/7\) | \(-77505793750000\) | \([]\) | \(972000\) | \(1.3472\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 338800.hq have rank \(1\).
Complex multiplication
The elliptic curves in class 338800.hq do not have complex multiplication.Modular form 338800.2.a.hq
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.