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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 177744cz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177744.o2 | 177744cz1 | \([0, -1, 0, 67536, -7743600]\) | \(224727548/299943\) | \(-45467984542030848\) | \([2]\) | \(1351680\) | \(1.8821\) | \(\Gamma_0(N)\)-optimal |
177744.o1 | 177744cz2 | \([0, -1, 0, -419144, -75489456]\) | \(26860713266/7394247\) | \(2241769324811433984\) | \([2]\) | \(2703360\) | \(2.2287\) |
Rank
sage: E.rank()
The elliptic curves in class 177744cz have rank \(1\).
Complex multiplication
The elliptic curves in class 177744cz do not have complex multiplication.Modular form 177744.2.a.cz
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.