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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 283920dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 283920.dk7 | 283920dk1 | \([0, -1, 0, -110920, 5040112]\) | \(7633736209/3870720\) | \(76526494238638080\) | \([2]\) | \(2654208\) | \(1.9322\) | \(\Gamma_0(N)\)-optimal |
| 283920.dk5 | 283920dk2 | \([0, -1, 0, -976200, -367376400]\) | \(5203798902289/57153600\) | \(1129961516492390400\) | \([2, 2]\) | \(5308416\) | \(2.2788\) | |
| 283920.dk4 | 283920dk3 | \([0, -1, 0, -7249480, 7515324400]\) | \(2131200347946769/2058000\) | \(40687914688512000\) | \([2]\) | \(7962624\) | \(2.4815\) | |
| 283920.dk6 | 283920dk4 | \([0, -1, 0, -219080, -923405328]\) | \(-58818484369/18600435000\) | \(-367741939965603840000\) | \([2]\) | \(10616832\) | \(2.6253\) | |
| 283920.dk2 | 283920dk5 | \([0, -1, 0, -15577800, -23659848720]\) | \(21145699168383889/2593080\) | \(51266772507525120\) | \([2]\) | \(10616832\) | \(2.6253\) | |
| 283920.dk3 | 283920dk6 | \([0, -1, 0, -7303560, 7397559792]\) | \(2179252305146449/66177562500\) | \(1308370756702464000000\) | \([2, 2]\) | \(15925248\) | \(2.8281\) | |
| 283920.dk8 | 283920dk7 | \([0, -1, 0, 1971160, 24893391600]\) | \(42841933504271/13565917968750\) | \(-268206468894000000000000\) | \([2]\) | \(31850496\) | \(3.1746\) | |
| 283920.dk1 | 283920dk8 | \([0, -1, 0, -17443560, -17636072208]\) | \(29689921233686449/10380965400750\) | \(205238014873717582848000\) | \([2]\) | \(31850496\) | \(3.1746\) |
Rank
sage: E.rank()
The elliptic curves in class 283920dk have rank \(2\).
Complex multiplication
The elliptic curves in class 283920dk do not have complex multiplication.Modular form 283920.2.a.dk
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.
