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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 138600by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
138600.u3 | 138600by1 | \([0, 0, 0, -115050, 14970125]\) | \(924093773824/3565485\) | \(649809641250000\) | \([2]\) | \(589824\) | \(1.7006\) | \(\Gamma_0(N)\)-optimal |
138600.u2 | 138600by2 | \([0, 0, 0, -170175, -850750]\) | \(186906097744/108056025\) | \(315091368900000000\) | \([2, 2]\) | \(1179648\) | \(2.0471\) | |
138600.u4 | 138600by3 | \([0, 0, 0, 680325, -6804250]\) | \(2985557859644/1729468125\) | \(-20172516210000000000\) | \([2]\) | \(2359296\) | \(2.3937\) | |
138600.u1 | 138600by4 | \([0, 0, 0, -1902675, -1007433250]\) | \(65308549273636/204604785\) | \(2386510212240000000\) | \([2]\) | \(2359296\) | \(2.3937\) |
Rank
sage: E.rank()
The elliptic curves in class 138600by have rank \(1\).
Complex multiplication
The elliptic curves in class 138600by do not have complex multiplication.Modular form 138600.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.