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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 22050dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.ej7 | 22050dz1 | \([1, -1, 1, -452255, -41077753]\) | \(7633736209/3870720\) | \(5187134998080000000\) | \([2]\) | \(442368\) | \(2.2836\) | \(\Gamma_0(N)\)-optimal |
| 22050.ej5 | 22050dz2 | \([1, -1, 1, -3980255, 3028282247]\) | \(5203798902289/57153600\) | \(76591290206025000000\) | \([2, 2]\) | \(884736\) | \(2.6301\) | |
| 22050.ej4 | 22050dz3 | \([1, -1, 1, -29558255, -61846345753]\) | \(2131200347946769/2058000\) | \(2757916828406250000\) | \([2]\) | \(1327104\) | \(2.8329\) | |
| 22050.ej6 | 22050dz4 | \([1, -1, 1, -893255, 7603216247]\) | \(-58818484369/18600435000\) | \(-24926361857228671875000\) | \([2]\) | \(1769472\) | \(2.9767\) | |
| 22050.ej2 | 22050dz5 | \([1, -1, 1, -63515255, 194850052247]\) | \(21145699168383889/2593080\) | \(3474975203791875000\) | \([2]\) | \(1769472\) | \(2.9767\) | |
| 22050.ej3 | 22050dz6 | \([1, -1, 1, -29778755, -60876586753]\) | \(2179252305146449/66177562500\) | \(88684263013438476562500\) | \([2, 2]\) | \(2654208\) | \(3.1794\) | |
| 22050.ej8 | 22050dz7 | \([1, -1, 1, 8036995, -204954594253]\) | \(42841933504271/13565917968750\) | \(-18179627530998229980468750\) | \([2]\) | \(5308416\) | \(3.5260\) | |
| 22050.ej1 | 22050dz8 | \([1, -1, 1, -71122505, 145263350747]\) | \(29689921233686449/10380965400750\) | \(13911486479024031105468750\) | \([2]\) | \(5308416\) | \(3.5260\) |
Rank
sage: E.rank()
The elliptic curves in class 22050dz have rank \(0\).
Complex multiplication
The elliptic curves in class 22050dz do not have complex multiplication.Modular form 22050.2.a.dz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
