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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 369600.dx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.dx1 | 369600dx3 | \([0, -1, 0, -142737488033, 20756598437147937]\) | \(78519570041710065450485106721/96428056919040\) | \(394969321140387840000000\) | \([2]\) | \(849346560\) | \(4.7019\) | |
369600.dx2 | 369600dx6 | \([0, -1, 0, -41981776033, -3030354502628063]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(768348885790912471311974400000000\) | \([2]\) | \(1698693120\) | \(5.0484\) | |
369600.dx3 | 369600dx4 | \([0, -1, 0, -9322576033, 293535577371937]\) | \(21876183941534093095979041/3572502915711058560000\) | \(14632971942752495861760000000000\) | \([2, 2]\) | \(849346560\) | \(4.7019\) | |
369600.dx4 | 369600dx2 | \([0, -1, 0, -8921168033, 324318352667937]\) | \(19170300594578891358373921/671785075055001600\) | \(2751631667425286553600000000\) | \([2, 2]\) | \(424673280\) | \(4.3553\) | |
369600.dx5 | 369600dx1 | \([0, -1, 0, -532560033, 5542860059937]\) | \(-4078208988807294650401/880065599546327040\) | \(-3604748695741755555840000000\) | \([2]\) | \(212336640\) | \(4.0087\) | \(\Gamma_0(N)\)-optimal |
369600.dx6 | 369600dx5 | \([0, -1, 0, 16914095967, 1647321615899937]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-1482002222352806246400000000000000\) | \([2]\) | \(1698693120\) | \(5.0484\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.dx have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.dx do not have complex multiplication.Modular form 369600.2.a.dx
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.