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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 107712bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
107712.ek3 | 107712bz1 | \([0, 0, 0, -222924, -40510352]\) | \(6411014266033/296208\) | \(56606230315008\) | \([2]\) | \(589824\) | \(1.7141\) | \(\Gamma_0(N)\)-optimal |
107712.ek2 | 107712bz2 | \([0, 0, 0, -234444, -36091280]\) | \(7457162887153/1370924676\) | \(261987785455435776\) | \([2, 2]\) | \(1179648\) | \(2.0607\) | |
107712.ek4 | 107712bz3 | \([0, 0, 0, 462516, -209773712]\) | \(57258048889007/132611470002\) | \(-25342446569116925952\) | \([2]\) | \(2359296\) | \(2.4073\) | |
107712.ek1 | 107712bz4 | \([0, 0, 0, -1115724, 420411760]\) | \(803760366578833/65593817586\) | \(12535173747885735936\) | \([2]\) | \(2359296\) | \(2.4073\) |
Rank
sage: E.rank()
The elliptic curves in class 107712bz have rank \(1\).
Complex multiplication
The elliptic curves in class 107712bz do not have complex multiplication.Modular form 107712.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.