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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 221760bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.jk5 | 221760bz1 | \([0, 0, 0, -191721612, 1197219428624]\) | \(-4078208988807294650401/880065599546327040\) | \(-168183155148527347213271040\) | \([2]\) | \(70778880\) | \(3.7533\) | \(\Gamma_0(N)\)-optimal |
| 221760.jk4 | 221760bz2 | \([0, 0, 0, -3211620492, 70052121852176]\) | \(19170300594578891358373921/671785075055001600\) | \(128380127075394169444761600\) | \([2, 2]\) | \(141557760\) | \(4.0999\) | |
| 221760.jk1 | 221760bz3 | \([0, 0, 0, -51385495692, 4483414985324816]\) | \(78519570041710065450485106721/96428056919040\) | \(18427688647125935063040\) | \([2]\) | \(283115520\) | \(4.4465\) | |
| 221760.jk3 | 221760bz4 | \([0, 0, 0, -3356127372, 63403013486864]\) | \(21876183941534093095979041/3572502915711058560000\) | \(682715938961060446926274560000\) | \([2, 2]\) | \(283115520\) | \(4.4465\) | |
| 221760.jk6 | 221760bz5 | \([0, 0, 0, 6089074548, 355822686849296]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-69144295686092528232038400000000\) | \([2]\) | \(566231040\) | \(4.7930\) | |
| 221760.jk2 | 221760bz6 | \([0, 0, 0, -15113439372, -654559595255536]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(35848085615460812261531477606400\) | \([2]\) | \(566231040\) | \(4.7930\) |
Rank
sage: E.rank()
The elliptic curves in class 221760bz have rank \(1\).
Complex multiplication
The elliptic curves in class 221760bz do not have complex multiplication.Modular form 221760.2.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.
