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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 100800hc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.o2 | 100800hc1 | \([0, 0, 0, -110640, 23558600]\) | \(-1605176213504/1640558367\) | \(-153083782341504000\) | \([2]\) | \(1032192\) | \(1.9930\) | \(\Gamma_0(N)\)-optimal |
100800.o1 | 100800hc2 | \([0, 0, 0, -2078940, 1153362800]\) | \(665567485783184/257298363\) | \(384144397572096000\) | \([2]\) | \(2064384\) | \(2.3395\) |
Rank
sage: E.rank()
The elliptic curves in class 100800hc have rank \(1\).
Complex multiplication
The elliptic curves in class 100800hc do not have complex multiplication.Modular form 100800.2.a.hc
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 Cremona 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 Cremona labels.