Normalized defining polynomial
\( x^{16} + 88 x^{14} - 44 x^{13} + 3622 x^{12} - 2744 x^{11} + 84906 x^{10} - 69432 x^{9} + 1203805 x^{8} - 853752 x^{7} + 10143918 x^{6} - 5250312 x^{5} + 47701212 x^{4} - 13773424 x^{3} + 101881698 x^{2} - 3534252 x + 60133751 \)
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: | \(43381651460325376000000000000=2^{32}\cdot 5^{12}\cdot 11^{4}\cdot 41^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.64$ | 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: | $2, 5, 11, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2711858544813982218888248098469698636592098131757069102} a^{15} + \frac{54961270859394965681824497540447812352305219609427062}{1355929272406991109444124049234849318296049065878534551} a^{14} - \frac{122411851087082420296126902033486115417814103350035356}{1355929272406991109444124049234849318296049065878534551} a^{13} + \frac{38861907981509289702023416500522658887319572915691104}{1355929272406991109444124049234849318296049065878534551} a^{12} - \frac{50394223004048973624040627254946217502095675369881417}{1355929272406991109444124049234849318296049065878534551} a^{11} + \frac{575449277007829022453676906335737540954650266587789317}{2711858544813982218888248098469698636592098131757069102} a^{10} + \frac{37089229196466018957849618170688562317723110419066441}{1355929272406991109444124049234849318296049065878534551} a^{9} + \frac{583240617005786148651828935436630301400791210378863381}{2711858544813982218888248098469698636592098131757069102} a^{8} + \frac{633746963699317864685674427010543276525048054384882818}{1355929272406991109444124049234849318296049065878534551} a^{7} + \frac{178675869212221242999155936461693885543720478798672015}{2711858544813982218888248098469698636592098131757069102} a^{6} + \frac{226295693136078972456377551061430696107938305919515572}{1355929272406991109444124049234849318296049065878534551} a^{5} + \frac{777757266056283583907690194069198783154512973255749475}{2711858544813982218888248098469698636592098131757069102} a^{4} + \frac{58712769199433493988508085293347407742213254888190291}{2711858544813982218888248098469698636592098131757069102} a^{3} + \frac{209593016471336033466035644551123862523614518440660185}{2711858544813982218888248098469698636592098131757069102} a^{2} - \frac{490953785353082510664397243723771880255494902795619882}{1355929272406991109444124049234849318296049065878534551} a - \frac{426919736290465320353221752800001975266547313793754489}{2711858544813982218888248098469698636592098131757069102}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{653696523608921840925218029830740612558834}{14575026307435060457741226572162497643753684963921} a^{15} + \frac{546083227580788435099919731710740601703906}{14575026307435060457741226572162497643753684963921} a^{14} - \frac{53914665619163155510026259083786681442723367}{14575026307435060457741226572162497643753684963921} a^{13} + \frac{71107911941524493769474261885100687344229297}{14575026307435060457741226572162497643753684963921} a^{12} - \frac{2102619057401436514928447099177732525930165486}{14575026307435060457741226572162497643753684963921} a^{11} + \frac{6302950754977708124397532462717149981311393413}{29150052614870120915482453144324995287507369927842} a^{10} - \frac{46034298902295288364011184823417864463081553736}{14575026307435060457741226572162497643753684963921} a^{9} + \frac{131855844065125519056499910355322174565503728543}{29150052614870120915482453144324995287507369927842} a^{8} - \frac{593329785118755923078556431931805174448824473710}{14575026307435060457741226572162497643753684963921} a^{7} + \frac{1377907355775121094103262154360977108364782823527}{29150052614870120915482453144324995287507369927842} a^{6} - \frac{4286476929227463481709251967395995095480030572992}{14575026307435060457741226572162497643753684963921} a^{5} + \frac{7026029145442012716063692620162268084090494424053}{29150052614870120915482453144324995287507369927842} a^{4} - \frac{15912078225144269198937759236189244754218628920818}{14575026307435060457741226572162497643753684963921} a^{3} + \frac{17318191347645791053928151701659048103441337400153}{29150052614870120915482453144324995287507369927842} a^{2} - \frac{19595729540807421806292512304834484377202682986537}{14575026307435060457741226572162497643753684963921} a + \frac{35340718765025632827590435716515988028541046253155}{29150052614870120915482453144324995287507369927842} \) (order $10$) | 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: | \( 28150374.3329 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3$ (as 16T203):
| A solvable group of order 128 |
| The 41 conjugacy class representatives for $C_2^4.C_2^3$ |
| Character table for $C_2^4.C_2^3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $11$ | 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.2.2 | $x^{4} - 11 x^{2} + 847$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.2 | $x^{4} - 11 x^{2} + 847$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 41 | Data not computed | ||||||