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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 154560.hc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
154560.hc1 | 154560z1 | \([0, 1, 0, -85, -967]\) | \(-1073741824/5325075\) | \(-340804800\) | \([]\) | \(62208\) | \(0.32167\) | \(\Gamma_0(N)\)-optimal |
154560.hc2 | 154560z2 | \([0, 1, 0, 755, 23225]\) | \(742692847616/3992296875\) | \(-255507000000\) | \([]\) | \(186624\) | \(0.87097\) |
Rank
sage: E.rank()
The elliptic curves in class 154560.hc have rank \(1\).
Complex multiplication
The elliptic curves in class 154560.hc do not have complex multiplication.Modular form 154560.2.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 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.