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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 166464ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 166464.fv5 | 166464ck1 | \([0, 0, 0, -5663244, 4810888208]\) | \(4354703137/352512\) | \(1626053700565365424128\) | \([2]\) | \(7077888\) | \(2.8130\) | \(\Gamma_0(N)\)-optimal |
| 166464.fv4 | 166464ck2 | \([0, 0, 0, -18980364, -26249962480]\) | \(163936758817/30338064\) | \(139942246604906761814016\) | \([2, 2]\) | \(14155776\) | \(3.1596\) | |
| 166464.fv6 | 166464ck3 | \([0, 0, 0, 37617396, -152870471152]\) | \(1276229915423/2927177028\) | \(-13502368823158724474437632\) | \([2]\) | \(28311552\) | \(3.5061\) | |
| 166464.fv2 | 166464ck4 | \([0, 0, 0, -288652044, -1887523897840]\) | \(576615941610337/27060804\) | \(124825028607463130136576\) | \([2, 2]\) | \(28311552\) | \(3.5061\) | |
| 166464.fv3 | 166464ck5 | \([0, 0, 0, -273670284, -2092192717552]\) | \(-491411892194497/125563633938\) | \(-579195067462440449563164672\) | \([2]\) | \(56623104\) | \(3.8527\) | |
| 166464.fv1 | 166464ck6 | \([0, 0, 0, -4618380684, -120804386941168]\) | \(2361739090258884097/5202\) | \(23995584122926399488\) | \([2]\) | \(56623104\) | \(3.8527\) |
Rank
sage: E.rank()
The elliptic curves in class 166464ck have rank \(1\).
Complex multiplication
The elliptic curves in class 166464ck do not have complex multiplication.Modular form 166464.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
