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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 137280.dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137280.dk1 | 137280cu4 | \([0, -1, 0, -102643425, -400228206975]\) | \(912446049969377120252018/17177299425\) | \(2251462990233600\) | \([2]\) | \(11010048\) | \(2.9339\) | |
| 137280.dk2 | 137280cu3 | \([0, -1, 0, -6987425, -5069891775]\) | \(287849398425814280018/81784533026485575\) | \(10719662312847517286400\) | \([2]\) | \(11010048\) | \(2.9339\) | |
| 137280.dk3 | 137280cu2 | \([0, -1, 0, -6415425, -6251529375]\) | \(445574312599094932036/61129333175625\) | \(4006171978997760000\) | \([2, 2]\) | \(5505024\) | \(2.5873\) | |
| 137280.dk4 | 137280cu1 | \([0, -1, 0, -365425, -115619375]\) | \(-329381898333928144/162600887109375\) | \(-2664052934400000000\) | \([2]\) | \(2752512\) | \(2.2408\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 137280.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 137280.dk do not have complex multiplication.Modular form 137280.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
