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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 80850.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.g1 | 80850j4 | \([1, 1, 0, -109283389275, -13905313004059875]\) | \(78519570041710065450485106721/96428056919040\) | \(177260382319814640000000\) | \([2]\) | \(212336640\) | \(4.6351\) | |
| 80850.g2 | 80850j6 | \([1, 1, 0, -32142297275, 2030096752528125]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(344831382997188802098779587500000\) | \([2]\) | \(424673280\) | \(4.9817\) | |
| 80850.g3 | 80850j3 | \([1, 1, 0, -7137597275, -196646796571875]\) | \(21876183941534093095979041/3572502915711058560000\) | \(6567209305163911383210000000000\) | \([2, 2]\) | \(212336640\) | \(4.6351\) | |
| 80850.g4 | 80850j2 | \([1, 1, 0, -6830269275, -217268812699875]\) | \(19170300594578891358373921/671785075055001600\) | \(1234919410861654425600000000\) | \([2, 2]\) | \(106168320\) | \(4.2885\) | |
| 80850.g5 | 80850j1 | \([1, 1, 0, -407741275, -3713334171875]\) | \(-4078208988807294650401/880065599546327040\) | \(-1617794339391028592640000000\) | \([2]\) | \(53084160\) | \(3.9420\) | \(\Gamma_0(N)\)-optimal |
| 80850.g6 | 80850j5 | \([1, 1, 0, 12949854725, -1103575166919875]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-665115659551945885020117187500000\) | \([2]\) | \(424673280\) | \(4.9817\) |
Rank
sage: E.rank()
The elliptic curves in class 80850.g have rank \(0\).
Complex multiplication
The elliptic curves in class 80850.g do not have complex multiplication.Modular form 80850.2.a.g
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.
