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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 44100by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44100.u2 | 44100by1 | \([0, 0, 0, -2100, 30625]\) | \(16384/3\) | \(187535250000\) | \([2]\) | \(49152\) | \(0.88167\) | \(\Gamma_0(N)\)-optimal |
44100.u1 | 44100by2 | \([0, 0, 0, -9975, -355250]\) | \(109744/9\) | \(9001692000000\) | \([2]\) | \(98304\) | \(1.2282\) |
Rank
sage: E.rank()
The elliptic curves in class 44100by have rank \(2\).
Complex multiplication
The elliptic curves in class 44100by do not have complex multiplication.Modular form 44100.2.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.