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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 308112.by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
308112.by1 | 308112by2 | \([0, 1, 0, -178003296, -914150637324]\) | \(1294373635812597347281/2083292441154\) | \(1003918427788603170816\) | \([]\) | \(33264000\) | \(3.2942\) | |
308112.by2 | 308112by1 | \([0, 1, 0, -1673856, 791709876]\) | \(1076291879750641/60150618144\) | \(28985999663200075776\) | \([]\) | \(6652800\) | \(2.4895\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 308112.by have rank \(1\).
Complex multiplication
The elliptic curves in class 308112.by do not have complex multiplication.Modular form 308112.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.