Normalized defining polynomial
\( x^{16} - 28 x^{14} - 12 x^{13} + 360 x^{12} + 72 x^{11} - 2630 x^{10} + 840 x^{9} + 13881 x^{8} - 16152 x^{7} - 37930 x^{6} + 48084 x^{5} + 76402 x^{4} - 60288 x^{3} + 51360 x^{2} + 169920 x + 67104 \)
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$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(601,·)$, $\chi_{840}(29,·)$, $\chi_{840}(419,·)$, $\chi_{840}(421,·)$, $\chi_{840}(811,·)$, $\chi_{840}(239,·)$, $\chi_{840}(449,·)$, $\chi_{840}(659,·)$, $\chi_{840}(181,·)$, $\chi_{840}(631,·)$, $\chi_{840}(839,·)$, $\chi_{840}(629,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{282001896} a^{14} + \frac{4404745}{94000632} a^{13} + \frac{233437}{35250237} a^{12} - \frac{6640289}{94000632} a^{11} - \frac{4065665}{23500158} a^{10} - \frac{4681773}{31333544} a^{9} - \frac{11480347}{141000948} a^{8} - \frac{45532459}{94000632} a^{7} + \frac{15020741}{31333544} a^{6} + \frac{2983643}{11750079} a^{5} + \frac{3972671}{12818268} a^{4} - \frac{18679789}{47000316} a^{3} - \frac{48495283}{141000948} a^{2} + \frac{4400323}{11750079} a - \frac{3299449}{11750079}$, $\frac{1}{30494865062221263889419283632} a^{15} - \frac{2780670274000860529}{5082477510370210648236547272} a^{14} - \frac{95604866167702688479201039}{1905929066388828993088705227} a^{13} + \frac{44661034944459494908740203}{5082477510370210648236547272} a^{12} + \frac{56773909600587226464489718}{635309688796276331029568409} a^{11} - \frac{219053328508055842872476183}{1694159170123403549412182424} a^{10} + \frac{3345972568840061387616168809}{15247432531110631944709641816} a^{9} - \frac{1136740203675529006001652605}{5082477510370210648236547272} a^{8} - \frac{1127120218530414154313299807}{3388318340246807098824364848} a^{7} + \frac{582289542865833799933067561}{1270619377592552662059136818} a^{6} - \frac{332286835353720189976085267}{15247432531110631944709641816} a^{5} + \frac{125903423245256377808296336}{635309688796276331029568409} a^{4} - \frac{5328977420753720825737924699}{15247432531110631944709641816} a^{3} + \frac{199186250660149947390385412}{635309688796276331029568409} a^{2} - \frac{570092259175386075916337359}{1270619377592552662059136818} a + \frac{26497722700540043095155755}{70589965421808481225507601}$
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: | \( -\frac{10543616044304042891987}{222590255928622364156345136} a^{15} + \frac{796489639419496098151}{37098375988103727359390856} a^{14} + \frac{18340608486856590348275}{13911890995538897759771571} a^{13} - \frac{400254038422019602207}{12366125329367909119796952} a^{12} - \frac{316904906518230710989607}{18549187994051863679695428} a^{11} + \frac{17887771018934314494915}{4122041776455969706598984} a^{10} + \frac{13689953011507807775446847}{111295127964311182078172568} a^{9} - \frac{107552556844166531833381}{1124193211760719010890632} a^{8} - \frac{15189278504665891013523191}{24732250658735818239593904} a^{7} + \frac{9701757723785650388867069}{9274593997025931839847714} a^{6} + \frac{145217304932825948940415879}{111295127964311182078172568} a^{5} - \frac{4460334812163675773436919}{1545765666170988639974619} a^{4} - \frac{235310957778679503992848735}{111295127964311182078172568} a^{3} + \frac{5706246730801751703791228}{1545765666170988639974619} a^{2} - \frac{20851969389496897443568388}{4637296998512965919923857} a - \frac{8007798635866528918351586}{1545765666170988639974619} \) (order $8$) | 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: | \( 968872.10852 \) (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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
| 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| $3$ | 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |