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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 6720by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.bu2 | 6720by1 | \([0, 1, 0, 104, -670]\) | \(1925134784/4465125\) | \(-285768000\) | \([2]\) | \(2304\) | \(0.30659\) | \(\Gamma_0(N)\)-optimal |
6720.bu1 | 6720by2 | \([0, 1, 0, -841, -8041]\) | \(16079333824/2953125\) | \(12096000000\) | \([2]\) | \(4608\) | \(0.65316\) |
Rank
sage: E.rank()
The elliptic curves in class 6720by have rank \(0\).
Complex multiplication
The elliptic curves in class 6720by do not have complex multiplication.Modular form 6720.2.a.by
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.