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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 21168ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21168.dg3 | 21168ck1 | \([0, 0, 0, 1029, -8918]\) | \(9261/8\) | \(-104088305664\) | \([]\) | \(18144\) | \(0.80159\) | \(\Gamma_0(N)\)-optimal |
| 21168.dg1 | 21168ck2 | \([0, 0, 0, -22491, -1315062]\) | \(-132651/2\) | \(-18970093707264\) | \([]\) | \(54432\) | \(1.3509\) | |
| 21168.dg2 | 21168ck3 | \([0, 0, 0, -10731, 567322]\) | \(-1167051/512\) | \(-59954864062464\) | \([]\) | \(54432\) | \(1.3509\) |
Rank
sage: E.rank()
The elliptic curves in class 21168ck have rank \(1\).
Complex multiplication
The elliptic curves in class 21168ck do not have complex multiplication.Modular form 21168.2.a.ck
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.
\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
