Normalized defining polynomial
\( x^{32} + 21 x^{28} + 4390 x^{24} + 27204 x^{20} + 1518849 x^{16} + 776784 x^{12} + 4625920 x^{8} + 147456 x^{4} + 65536 \)
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: | \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.29$ | 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: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(517,·)$, $\chi_{840}(391,·)$, $\chi_{840}(13,·)$, $\chi_{840}(659,·)$, $\chi_{840}(407,·)$, $\chi_{840}(29,·)$, $\chi_{840}(799,·)$, $\chi_{840}(419,·)$, $\chi_{840}(167,·)$, $\chi_{840}(169,·)$, $\chi_{840}(43,·)$, $\chi_{840}(559,·)$, $\chi_{840}(307,·)$, $\chi_{840}(769,·)$, $\chi_{840}(701,·)$, $\chi_{840}(629,·)$, $\chi_{840}(713,·)$, $\chi_{840}(631,·)$, $\chi_{840}(461,·)$, $\chi_{840}(547,·)$, $\chi_{840}(601,·)$, $\chi_{840}(743,·)$, $\chi_{840}(617,·)$, $\chi_{840}(491,·)$, $\chi_{840}(113,·)$, $\chi_{840}(757,·)$, $\chi_{840}(503,·)$, $\chi_{840}(377,·)$, $\chi_{840}(251,·)$, $\chi_{840}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{2} a^{13} + \frac{1}{3} a^{9} + \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{1}{4} a^{14} + \frac{1}{6} a^{10} + \frac{1}{12} a^{2}$, $\frac{1}{24} a^{19} - \frac{1}{8} a^{15} - \frac{5}{12} a^{11} - \frac{1}{2} a^{7} + \frac{1}{24} a^{3}$, $\frac{1}{48} a^{20} + \frac{5}{48} a^{16} - \frac{5}{24} a^{12} + \frac{1}{12} a^{8} + \frac{1}{48} a^{4} - \frac{1}{3}$, $\frac{1}{96} a^{21} + \frac{5}{96} a^{17} - \frac{5}{48} a^{13} + \frac{1}{24} a^{9} + \frac{1}{96} a^{5} + \frac{1}{3} a$, $\frac{1}{192} a^{22} + \frac{5}{192} a^{18} + \frac{43}{96} a^{14} - \frac{23}{48} a^{10} + \frac{1}{192} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{384} a^{23} + \frac{5}{384} a^{19} + \frac{43}{192} a^{15} + \frac{25}{96} a^{11} - \frac{191}{384} a^{7} - \frac{1}{6} a^{3}$, $\frac{1}{44833536} a^{24} + \frac{75415}{14944512} a^{20} - \frac{825191}{7472256} a^{16} - \frac{315055}{1245376} a^{12} + \frac{55969}{1358592} a^{8} + \frac{998}{58377} a^{4} + \frac{27691}{175131}$, $\frac{1}{89667072} a^{25} + \frac{75415}{29889024} a^{21} - \frac{825191}{14944512} a^{17} - \frac{315055}{2490752} a^{13} - \frac{1302623}{2717184} a^{9} + \frac{499}{58377} a^{5} - \frac{73720}{175131} a$, $\frac{1}{179334144} a^{26} + \frac{75415}{59778048} a^{22} - \frac{825191}{29889024} a^{18} - \frac{315055}{4981504} a^{14} + \frac{1414561}{5434368} a^{10} + \frac{499}{116754} a^{6} + \frac{101411}{350262} a^{2}$, $\frac{1}{358668288} a^{27} + \frac{75415}{119556096} a^{23} - \frac{825191}{59778048} a^{19} - \frac{315055}{9963008} a^{15} - \frac{4019807}{10868736} a^{11} + \frac{499}{233508} a^{7} - \frac{248851}{700524} a^{3}$, $\frac{1}{6228156500240043257856} a^{28} - \frac{68568441886267}{6228156500240043257856} a^{24} - \frac{7137685493475178343}{1038026083373340542976} a^{20} - \frac{40245180515947037069}{519013041686670271488} a^{16} - \frac{638453471338729955861}{2076052166746681085952} a^{12} - \frac{79983723759694214}{168949557840712979} a^{8} - \frac{23607470955479135}{99707935774847004} a^{4} - \frac{526758900987774898}{1520546020566416811}$, $\frac{1}{12456313000480086515712} a^{29} - \frac{68568441886267}{12456313000480086515712} a^{25} - \frac{7137685493475178343}{2076052166746681085952} a^{21} - \frac{40245180515947037069}{1038026083373340542976} a^{17} + \frac{1437598695407951130091}{4152104333493362171904} a^{13} + \frac{88965834081018765}{337899115681425958} a^{9} - \frac{23607470955479135}{199415871549694008} a^{5} + \frac{993787119578641913}{3041092041132833622} a$, $\frac{1}{24912626000960173031424} a^{30} - \frac{68568441886267}{24912626000960173031424} a^{26} - \frac{7137685493475178343}{4152104333493362171904} a^{22} - \frac{40245180515947037069}{2076052166746681085952} a^{18} - \frac{2714505638085411041813}{8304208666986724343808} a^{14} + \frac{88965834081018765}{675798231362851916} a^{10} - \frac{23607470955479135}{398831743099388016} a^{6} - \frac{2047304921554191709}{6082184082265667244} a^{2}$, $\frac{1}{49825252001920346062848} a^{31} - \frac{68568441886267}{49825252001920346062848} a^{27} - \frac{7137685493475178343}{8304208666986724343808} a^{23} - \frac{40245180515947037069}{4152104333493362171904} a^{19} - \frac{2714505638085411041813}{16608417333973448687616} a^{15} - \frac{586832397281833151}{1351596462725703832} a^{11} - \frac{23607470955479135}{797663486198776032} a^{7} + \frac{4034879160711475535}{12164368164531334488} a^{3}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{40}$, which has order $1280$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{6569874683}{1917269941542912} a^{30} - \frac{15391798375}{213029993504768} a^{26} - \frac{4808954884775}{319544990257152} a^{22} - \frac{5032933433855}{53257498376192} a^{18} - \frac{302876741913275}{58099089137664} a^{14} - \frac{248357900466725}{79886247564288} a^{10} - \frac{243894337122145}{14978671418304} a^{6} - \frac{69558908725}{28368695868} a^{2} \) (order $4$) | 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: | \( 12132335683441.611 \) (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 | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{32}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 | Data not computed | ||||||
| $3$ | 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 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |