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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 44880cy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 44880.cu4 | 44880cy1 | \([0, 1, 0, -116960, -15434892]\) | \(43199583152847841/89760000\) | \(367656960000\) | \([2]\) | \(147456\) | \(1.4686\) | \(\Gamma_0(N)\)-optimal |
| 44880.cu3 | 44880cy2 | \([0, 1, 0, -118240, -15081100]\) | \(44633474953947361/1967006250000\) | \(8056857600000000\) | \([2, 2]\) | \(294912\) | \(1.8152\) | |
| 44880.cu5 | 44880cy3 | \([0, 1, 0, 61280, -56657932]\) | \(6213165856218719/342407226562500\) | \(-1402500000000000000\) | \([2]\) | \(589824\) | \(2.1617\) | |
| 44880.cu2 | 44880cy4 | \([0, 1, 0, -318240, 49158900]\) | \(870220733067747361/247623269602500\) | \(1014264912291840000\) | \([2, 4]\) | \(589824\) | \(2.1617\) | |
| 44880.cu6 | 44880cy5 | \([0, 1, 0, 837760, 325211700]\) | \(15875306080318016639/20322604533582450\) | \(-83241388169553715200\) | \([4]\) | \(1179648\) | \(2.5083\) | |
| 44880.cu1 | 44880cy6 | \([0, 1, 0, -4674240, 3887666100]\) | \(2757381641970898311361/379829992662450\) | \(1555783649945395200\) | \([4]\) | \(1179648\) | \(2.5083\) |
Rank
sage: E.rank()
The elliptic curves in class 44880cy have rank \(0\).
Complex multiplication
The elliptic curves in class 44880cy do not have complex multiplication.Modular form 44880.2.a.cy
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.
