Normalized defining polynomial
\( x^{16} + 984 x^{14} + 373428 x^{12} + 70440624 x^{10} + 7238830194 x^{8} + 414992585088 x^{6} + 12938231720448 x^{4} + 200034687172608 x^{2} + 1185571438608384 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1147634942730979008947338773912344095761629184=2^{62}\cdot 3^{8}\cdot 41^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $654.99$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3936=2^{5}\cdot 3\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3936}(1,·)$, $\chi_{3936}(1667,·)$, $\chi_{3936}(73,·)$, $\chi_{3936}(3851,·)$, $\chi_{3936}(1859,·)$, $\chi_{3936}(3289,·)$, $\chi_{3936}(1883,·)$, $\chi_{3936}(3361,·)$, $\chi_{3936}(3611,·)$, $\chi_{3936}(1393,·)$, $\chi_{3936}(1321,·)$, $\chi_{3936}(1643,·)$, $\chi_{3936}(1969,·)$, $\chi_{3936}(3635,·)$, $\chi_{3936}(3827,·)$, $\chi_{3936}(2041,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{27} a^{5} - \frac{1}{3} a$, $\frac{1}{81} a^{6} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{7} - \frac{1}{9} a^{3}$, $\frac{1}{19926} a^{8} - \frac{1}{27} a^{4}$, $\frac{1}{119556} a^{9} - \frac{1}{81} a^{5} + \frac{1}{18} a$, $\frac{1}{4304016} a^{10} - \frac{1}{59778} a^{8} + \frac{5}{2916} a^{6} + \frac{4}{81} a^{4} + \frac{97}{648} a^{2}$, $\frac{1}{8608032} a^{11} + \frac{5}{5832} a^{7} - \frac{1}{162} a^{5} + \frac{97}{1296} a^{3} - \frac{5}{18} a$, $\frac{1}{309889152} a^{12} + \frac{1}{12912048} a^{10} + \frac{85}{8608032} a^{8} + \frac{97}{17496} a^{6} - \frac{2303}{46656} a^{4} - \frac{28}{243} a^{2}$, $\frac{1}{1239556608} a^{13} + \frac{1}{51648192} a^{11} - \frac{59}{34432128} a^{9} + \frac{313}{69984} a^{7} + \frac{577}{186624} a^{5} - \frac{34}{243} a^{3} - \frac{1}{9} a$, $\frac{1}{27017117847482641007616} a^{14} - \frac{11037165119}{27456420576710001024} a^{12} - \frac{745507556819}{18304280384473334016} a^{10} + \frac{33494567723657}{1525356698706111168} a^{8} - \frac{21030954866941759}{4067617863216296448} a^{6} - \frac{14797265062231}{516719748884184} a^{4} - \frac{18033195162313}{172239916294728} a^{2} - \frac{20578978631}{88600779987}$, $\frac{1}{972616242509375076274176} a^{15} - \frac{34604938135}{123553892595195004608} a^{13} - \frac{34768207071827}{658954093841040024576} a^{11} + \frac{166927342384079}{54912841153420002048} a^{9} - \frac{790335224950545151}{146434243075786672128} a^{7} + \frac{2279956392970255}{148815287678644992} a^{5} - \frac{22156099599869}{775079623326276} a^{3} + \frac{194207010313}{797407019883} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{3495176}$, which has order $3579060224$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 11418873.067779342 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_8$ (as 16T5):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_8\times C_2$ |
| Character table for $C_8\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.31.1 | $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{2} + 50$ | $8$ | $1$ | $31$ | $C_8$ | $[3, 4, 5]$ |
| 2.8.31.1 | $x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{2} + 50$ | $8$ | $1$ | $31$ | $C_8$ | $[3, 4, 5]$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.8.7.2 | $x^{8} - 1476$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.2 | $x^{8} - 1476$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |