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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 169050gt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.ee4 | 169050gt1 | \([1, 0, 1, -931026, -345793052]\) | \(48551226272641/9273600\) | \(17047340100000000\) | \([2]\) | \(2359296\) | \(2.1154\) | \(\Gamma_0(N)\)-optimal |
169050.ee3 | 169050gt2 | \([1, 0, 1, -1029026, -268569052]\) | \(65553197996161/20996010000\) | \(38596243445156250000\) | \([2, 2]\) | \(4718592\) | \(2.4620\) | |
169050.ee2 | 169050gt3 | \([1, 0, 1, -6541526, 6236180948]\) | \(16840406336564161/604708416900\) | \(1111614695935439062500\) | \([2, 2]\) | \(9437184\) | \(2.8085\) | |
169050.ee6 | 169050gt4 | \([1, 0, 1, 2915474, -1830591052]\) | \(1490881681033919/1650501562500\) | \(-3034060286352539062500\) | \([2]\) | \(9437184\) | \(2.8085\) | |
169050.ee1 | 169050gt5 | \([1, 0, 1, -103745276, 406715630948]\) | \(67176973097223766561/91487391870\) | \(168178127595525468750\) | \([2]\) | \(18874368\) | \(3.1551\) | |
169050.ee5 | 169050gt6 | \([1, 0, 1, 2462224, 22082780948]\) | \(898045580910239/115117148363070\) | \(-211615896683856600468750\) | \([2]\) | \(18874368\) | \(3.1551\) |
Rank
sage: E.rank()
The elliptic curves in class 169050gt have rank \(0\).
Complex multiplication
The elliptic curves in class 169050gt do not have complex multiplication.Modular form 169050.2.a.gt
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.