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SageMath
E = EllipticCurve("rr1")
E.isogeny_class()
Elliptic curves in class 369600.rr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.rr1 | 369600rr4 | \([0, 1, 0, -142737488033, -20756598437147937]\) | \(78519570041710065450485106721/96428056919040\) | \(394969321140387840000000\) | \([2]\) | \(849346560\) | \(4.7019\) | |
369600.rr2 | 369600rr5 | \([0, 1, 0, -41981776033, 3030354502628063]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(768348885790912471311974400000000\) | \([2]\) | \(1698693120\) | \(5.0484\) | |
369600.rr3 | 369600rr3 | \([0, 1, 0, -9322576033, -293535577371937]\) | \(21876183941534093095979041/3572502915711058560000\) | \(14632971942752495861760000000000\) | \([2, 2]\) | \(849346560\) | \(4.7019\) | |
369600.rr4 | 369600rr2 | \([0, 1, 0, -8921168033, -324318352667937]\) | \(19170300594578891358373921/671785075055001600\) | \(2751631667425286553600000000\) | \([2, 2]\) | \(424673280\) | \(4.3553\) | |
369600.rr5 | 369600rr1 | \([0, 1, 0, -532560033, -5542860059937]\) | \(-4078208988807294650401/880065599546327040\) | \(-3604748695741755555840000000\) | \([2]\) | \(212336640\) | \(4.0087\) | \(\Gamma_0(N)\)-optimal |
369600.rr6 | 369600rr6 | \([0, 1, 0, 16914095967, -1647321615899937]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-1482002222352806246400000000000000\) | \([2]\) | \(1698693120\) | \(5.0484\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.rr have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.rr do not have complex multiplication.Modular form 369600.2.a.rr
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.