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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 98736bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
98736.n5 | 98736bz1 | \([0, -1, 0, -65864, 6055920]\) | \(4354703137/352512\) | \(2557937710006272\) | \([2]\) | \(491520\) | \(1.6995\) | \(\Gamma_0(N)\)-optimal |
98736.n4 | 98736bz2 | \([0, -1, 0, -220744, -32849936]\) | \(163936758817/30338064\) | \(220142514167414784\) | \([2, 2]\) | \(983040\) | \(2.0460\) | |
98736.n6 | 98736bz3 | \([0, -1, 0, 437496, -191880720]\) | \(1276229915423/2927177028\) | \(-21240515227241299968\) | \([2]\) | \(1966080\) | \(2.3926\) | |
98736.n2 | 98736bz4 | \([0, -1, 0, -3357064, -2366272016]\) | \(576615941610337/27060804\) | \(196361687019700224\) | \([2, 2]\) | \(1966080\) | \(2.3926\) | |
98736.n3 | 98736bz5 | \([0, -1, 0, -3182824, -2623032080]\) | \(-491411892194497/125563633938\) | \(-911129136754021244928\) | \([2]\) | \(3932160\) | \(2.7392\) | |
98736.n1 | 98736bz6 | \([0, -1, 0, -53712424, -151498706192]\) | \(2361739090258884097/5202\) | \(37747344678912\) | \([2]\) | \(3932160\) | \(2.7392\) |
Rank
sage: E.rank()
The elliptic curves in class 98736bz have rank \(1\).
Complex multiplication
The elliptic curves in class 98736bz do not have complex multiplication.Modular form 98736.2.a.bz
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.