Normalized defining polynomial
\( x^{16} - 4 x^{15} + 34 x^{14} - 64 x^{13} + 539 x^{12} - 1000 x^{11} + 4842 x^{10} - 4648 x^{9} + 19562 x^{8} - 22604 x^{7} + 45816 x^{6} - 21740 x^{5} + 19433 x^{4} + 3616 x^{3} + 4684 x^{2} - 68 x + 1 \)
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: | \(35074927889488281600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.50$ | 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: | $2, 3, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1560=2^{3}\cdot 3\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1560}(1,·)$, $\chi_{1560}(389,·)$, $\chi_{1560}(961,·)$, $\chi_{1560}(1481,·)$, $\chi_{1560}(781,·)$, $\chi_{1560}(209,·)$, $\chi_{1560}(1429,·)$, $\chi_{1560}(989,·)$, $\chi_{1560}(1249,·)$, $\chi_{1560}(1301,·)$, $\chi_{1560}(1169,·)$, $\chi_{1560}(649,·)$, $\chi_{1560}(181,·)$, $\chi_{1560}(521,·)$, $\chi_{1560}(701,·)$, $\chi_{1560}(469,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{22} a^{11} - \frac{1}{11} a^{10} + \frac{1}{22} a^{9} - \frac{3}{22} a^{8} - \frac{1}{11} a^{7} - \frac{2}{11} a^{6} + \frac{1}{22} a^{5} + \frac{4}{11} a^{4} + \frac{7}{22} a^{3} + \frac{1}{22} a^{2} - \frac{2}{11} a - \frac{2}{11}$, $\frac{1}{132} a^{12} + \frac{2}{33} a^{10} - \frac{1}{11} a^{9} - \frac{5}{22} a^{8} - \frac{5}{22} a^{7} + \frac{1}{33} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} + \frac{4}{11} a^{3} - \frac{23}{66} a^{2} + \frac{9}{22} a - \frac{41}{132}$, $\frac{1}{10428} a^{13} + \frac{5}{3476} a^{12} - \frac{23}{5214} a^{11} - \frac{283}{1738} a^{10} + \frac{29}{158} a^{9} - \frac{175}{869} a^{8} + \frac{95}{5214} a^{7} + \frac{7}{79} a^{6} + \frac{139}{1738} a^{5} - \frac{221}{1738} a^{4} + \frac{1633}{5214} a^{3} + \frac{124}{869} a^{2} - \frac{1127}{10428} a - \frac{573}{3476}$, $\frac{1}{1251360} a^{14} - \frac{1}{625680} a^{13} + \frac{1279}{1251360} a^{12} + \frac{5459}{312840} a^{11} - \frac{133999}{625680} a^{10} + \frac{3277}{52140} a^{9} + \frac{62179}{312840} a^{8} - \frac{38623}{156420} a^{7} - \frac{154061}{625680} a^{6} - \frac{23951}{52140} a^{5} + \frac{6515}{125136} a^{4} - \frac{22337}{156420} a^{3} - \frac{38699}{113760} a^{2} + \frac{91907}{625680} a - \frac{470689}{1251360}$, $\frac{1}{4106985124657242991680} a^{15} + \frac{30100510475587}{456331680517471443520} a^{14} + \frac{53549159287870703}{1368995041552414330560} a^{13} - \frac{15218102674049598809}{4106985124657242991680} a^{12} + \frac{32299147398496752031}{2053492562328621495840} a^{11} - \frac{285203052913617336191}{2053492562328621495840} a^{10} - \frac{86775343701946869491}{1026746281164310747920} a^{9} - \frac{15445442944505525857}{342248760388103582640} a^{8} + \frac{369062912944675928399}{2053492562328621495840} a^{7} + \frac{170131260595383904643}{2053492562328621495840} a^{6} - \frac{45365976410936263153}{410698512465724299168} a^{5} + \frac{42721225089041176669}{684497520776207165280} a^{4} + \frac{17285079267833751159}{456331680517471443520} a^{3} + \frac{786413070298784741729}{4106985124657242991680} a^{2} + \frac{1730202161689946385781}{4106985124657242991680} a - \frac{220407862784879972113}{821397024931448598336}$
Class group and class number
$C_{4}\times C_{8}\times C_{8}$, which has order $256$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{20168794675795241071}{1368995041552414330560} a^{15} + \frac{80619917241805505317}{1368995041552414330560} a^{14} - \frac{685448422138966675559}{1368995041552414330560} a^{13} + \frac{39047467353144338333}{41484698228861040320} a^{12} - \frac{164617875508714929957}{20742349114430520160} a^{11} + \frac{10065981098241076423051}{684497520776207165280} a^{10} - \frac{24390061756432946103559}{342248760388103582640} a^{9} + \frac{23341715167352808560011}{342248760388103582640} a^{8} - \frac{65646880158665287059923}{228165840258735721760} a^{7} + \frac{227069662204185782750657}{684497520776207165280} a^{6} - \frac{92120066253267361137977}{136899504155241433056} a^{5} + \frac{216509592421739625361453}{684497520776207165280} a^{4} - \frac{386268058160253548027761}{1368995041552414330560} a^{3} - \frac{26197869414296511656243}{456331680517471443520} a^{2} - \frac{31025030070891296492357}{456331680517471443520} a + \frac{270243580583667117005}{273799008310482866112} \) (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: | \( 57386.4139318 \) (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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| $3$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |