Normalized defining polynomial
\( x^{32} - 89 x^{28} + 6657 x^{24} - 112318 x^{20} + 1589774 x^{16} - 112318 x^{12} + 6657 x^{8} - 89 x^{4} + 1 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(269546632883105820058091127981737482106510848676862099456=2^{64}\cdot 3^{16}\cdot 17^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $58.00$ | 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: | $2, 3, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(408=2^{3}\cdot 3\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(259,·)$, $\chi_{408}(137,·)$, $\chi_{408}(395,·)$, $\chi_{408}(13,·)$, $\chi_{408}(271,·)$, $\chi_{408}(149,·)$, $\chi_{408}(407,·)$, $\chi_{408}(157,·)$, $\chi_{408}(35,·)$, $\chi_{408}(293,·)$, $\chi_{408}(169,·)$, $\chi_{408}(47,·)$, $\chi_{408}(305,·)$, $\chi_{408}(307,·)$, $\chi_{408}(55,·)$, $\chi_{408}(191,·)$, $\chi_{408}(67,·)$, $\chi_{408}(203,·)$, $\chi_{408}(205,·)$, $\chi_{408}(341,·)$, $\chi_{408}(217,·)$, $\chi_{408}(353,·)$, $\chi_{408}(251,·)$, $\chi_{408}(101,·)$, $\chi_{408}(103,·)$, $\chi_{408}(361,·)$, $\chi_{408}(239,·)$, $\chi_{408}(115,·)$, $\chi_{408}(373,·)$, $\chi_{408}(89,·)$, $\chi_{408}(319,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{208} a^{12} - \frac{103}{208}$, $\frac{1}{208} a^{13} - \frac{103}{208} a$, $\frac{1}{208} a^{14} - \frac{103}{208} a^{2}$, $\frac{1}{208} a^{15} - \frac{103}{208} a^{3}$, $\frac{1}{208} a^{16} - \frac{103}{208} a^{4}$, $\frac{1}{208} a^{17} - \frac{103}{208} a^{5}$, $\frac{1}{208} a^{18} - \frac{103}{208} a^{6}$, $\frac{1}{208} a^{19} - \frac{103}{208} a^{7}$, $\frac{1}{208} a^{20} - \frac{103}{208} a^{8}$, $\frac{1}{208} a^{21} - \frac{103}{208} a^{9}$, $\frac{1}{208} a^{22} - \frac{103}{208} a^{10}$, $\frac{1}{208} a^{23} - \frac{103}{208} a^{11}$, $\frac{1}{1644032} a^{24} + \frac{7}{3952} a^{20} + \frac{9}{3952} a^{16} - \frac{207}{822016} a^{12} + \frac{1567}{3952} a^{8} - \frac{1551}{3952} a^{4} - \frac{703455}{1644032}$, $\frac{1}{1644032} a^{25} + \frac{7}{3952} a^{21} + \frac{9}{3952} a^{17} - \frac{207}{822016} a^{13} + \frac{1567}{3952} a^{9} - \frac{1551}{3952} a^{5} - \frac{703455}{1644032} a$, $\frac{1}{1644032} a^{26} + \frac{7}{3952} a^{22} + \frac{9}{3952} a^{18} - \frac{207}{822016} a^{14} + \frac{1567}{3952} a^{10} - \frac{1551}{3952} a^{6} - \frac{703455}{1644032} a^{2}$, $\frac{1}{1644032} a^{27} + \frac{7}{3952} a^{23} + \frac{9}{3952} a^{19} - \frac{207}{822016} a^{15} + \frac{1567}{3952} a^{11} - \frac{1551}{3952} a^{7} - \frac{703455}{1644032} a^{3}$, $\frac{1}{12545608192} a^{28} - \frac{675}{12545608192} a^{24} + \frac{1795}{15078856} a^{20} + \frac{5208113}{6272804096} a^{16} - \frac{13166659}{6272804096} a^{12} + \frac{2417819}{15078856} a^{8} - \frac{5572727263}{12545608192} a^{4} - \frac{2049281571}{12545608192}$, $\frac{1}{12545608192} a^{29} - \frac{675}{12545608192} a^{25} + \frac{1795}{15078856} a^{21} + \frac{5208113}{6272804096} a^{17} - \frac{13166659}{6272804096} a^{13} + \frac{2417819}{15078856} a^{9} - \frac{5572727263}{12545608192} a^{5} - \frac{2049281571}{12545608192} a$, $\frac{1}{12545608192} a^{30} - \frac{675}{12545608192} a^{26} + \frac{1795}{15078856} a^{22} + \frac{5208113}{6272804096} a^{18} - \frac{13166659}{6272804096} a^{14} + \frac{2417819}{15078856} a^{10} - \frac{5572727263}{12545608192} a^{6} - \frac{2049281571}{12545608192} a^{2}$, $\frac{1}{12545608192} a^{31} - \frac{675}{12545608192} a^{27} + \frac{1795}{15078856} a^{23} + \frac{5208113}{6272804096} a^{19} - \frac{13166659}{6272804096} a^{15} + \frac{2417819}{15078856} a^{11} - \frac{5572727263}{12545608192} a^{7} - \frac{2049281571}{12545608192} a^{3}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{20}$, which has order $2560$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{220250815}{12545608192} a^{29} + \frac{19586814385}{12545608192} a^{25} - \frac{3521225055}{30157712} a^{21} + \frac{12317454341265}{6272804096} a^{17} - \frac{174204163414143}{6272804096} a^{13} + \frac{248130665}{30157712} a^{9} - \frac{1380052575}{12545608192} a^{5} - \frac{37395278575}{12545608192} a \) (order $24$) | 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: | \( 9699756571026.293 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3\times C_4$ (as 32T34):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^3\times C_4$ |
| Character table for $C_2^3\times C_4$ is not computed |
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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |