Normalized defining polynomial
\( x^{16} - 4 x^{15} + 30 x^{14} - 56 x^{13} + 383 x^{12} - 660 x^{11} + 2958 x^{10} - 2628 x^{9} + 10176 x^{8} - 7192 x^{7} + 25000 x^{6} - 8448 x^{5} + 18363 x^{4} - 11064 x^{3} + 9876 x^{2} - 3100 x + 961 \)
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: | \(63456228123711897600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.99$ | 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: | $2, 3, 5, 7$ | 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$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(589,·)$, $\chi_{840}(281,·)$, $\chi_{840}(29,·)$, $\chi_{840}(671,·)$, $\chi_{840}(419,·)$, $\chi_{840}(421,·)$, $\chi_{840}(839,·)$, $\chi_{840}(169,·)$, $\chi_{840}(811,·)$, $\chi_{840}(559,·)$, $\chi_{840}(449,·)$, $\chi_{840}(251,·)$, $\chi_{840}(701,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{2}{9} a^{5} - \frac{2}{9} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{9} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{9} a^{10} + \frac{1}{3} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{27} a^{9} - \frac{8}{27} a^{5} - \frac{5}{27} a^{4} - \frac{13}{27} a^{3} + \frac{1}{27} a^{2} + \frac{7}{27} a - \frac{10}{27}$, $\frac{1}{162} a^{12} + \frac{2}{81} a^{10} + \frac{2}{81} a^{9} - \frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{11}{81} a^{6} - \frac{8}{27} a^{5} + \frac{23}{81} a^{4} - \frac{35}{81} a^{3} - \frac{10}{27} a^{2} - \frac{25}{81} a - \frac{71}{162}$, $\frac{1}{5022} a^{13} - \frac{2}{837} a^{12} - \frac{10}{2511} a^{11} + \frac{29}{2511} a^{10} + \frac{34}{837} a^{9} - \frac{16}{837} a^{8} - \frac{412}{2511} a^{7} - \frac{4}{27} a^{6} + \frac{326}{2511} a^{5} + \frac{244}{2511} a^{4} - \frac{94}{837} a^{3} - \frac{82}{2511} a^{2} + \frac{649}{5022} a + \frac{8}{27}$, $\frac{1}{426870} a^{14} - \frac{2}{23715} a^{13} - \frac{259}{426870} a^{12} + \frac{1664}{213435} a^{11} + \frac{7838}{213435} a^{10} - \frac{431}{42687} a^{9} + \frac{6506}{213435} a^{8} - \frac{421}{2635} a^{7} - \frac{4076}{213435} a^{6} + \frac{91837}{213435} a^{5} - \frac{2848}{6885} a^{4} - \frac{5095}{14229} a^{3} - \frac{143471}{426870} a^{2} - \frac{33766}{213435} a - \frac{2069}{13770}$, $\frac{1}{606016799331688530} a^{15} - \frac{91756714553}{121203359866337706} a^{14} - \frac{805263401569}{60601679933168853} a^{13} - \frac{79185337612253}{35648047019511090} a^{12} + \frac{608662068176678}{33667599962871585} a^{11} + \frac{1100721073664}{27679583416995} a^{10} + \frac{128043691835357}{3258154835116605} a^{9} + \frac{2442067901359235}{60601679933168853} a^{8} + \frac{8570628551240233}{303008399665844265} a^{7} + \frac{5618322227070512}{101002799888614755} a^{6} - \frac{135112577391039866}{303008399665844265} a^{5} - \frac{55495594108653923}{303008399665844265} a^{4} + \frac{211494019446453799}{606016799331688530} a^{3} - \frac{2203466260801133}{11882682339837030} a^{2} - \frac{12798451023124522}{33667599962871585} a + \frac{3517657495686131}{19548929010699630}$
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{8}$, which has order $128$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{3861412501099}{11222533320957195} a^{15} + \frac{31541961598273}{22445066641914390} a^{14} - \frac{117299073047204}{11222533320957195} a^{13} + \frac{75225462528626}{3740844440319065} a^{12} - \frac{499278631710878}{3740844440319065} a^{11} + \frac{145431162928}{615101853711} a^{10} - \frac{11641008974853658}{11222533320957195} a^{9} + \frac{10860575738681723}{11222533320957195} a^{8} - \frac{39960507777897272}{11222533320957195} a^{7} + \frac{28797204720210764}{11222533320957195} a^{6} - \frac{5763258366420578}{660149018879835} a^{5} + \frac{2230736192907064}{748168888063813} a^{4} - \frac{23686462219941447}{3740844440319065} a^{3} + \frac{18380057540777517}{7481688880638130} a^{2} - \frac{37954254801358714}{11222533320957195} a + \frac{25598114616902}{24134480260123} \) (order $6$) | 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: | \( 114559.255041 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_2^4$ |
| Character table for $C_2^4$ |
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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $3$ | 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 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |