Normalized defining polynomial
\( x^{16} + 44 x^{14} - 12 x^{13} + 840 x^{12} - 144 x^{11} + 8542 x^{10} + 1452 x^{9} + 48381 x^{8} + 34392 x^{7} + 164762 x^{6} + 204360 x^{5} + 422866 x^{4} + 517656 x^{3} + 817764 x^{2} + 628056 x + 736164 \)
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$$\chi_{840}(1,·)$, $\chi_{840}(839,·)$, $\chi_{840}(589,·)$, $\chi_{840}(461,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(251,·)$, $\chi_{840}(41,·)$, $\chi_{840}(799,·)$, $\chi_{840}(419,·)$, $\chi_{840}(421,·)$, $\chi_{840}(169,·)$, $\chi_{840}(629,·)$, $\chi_{840}(631,·)$, $\chi_{840}(379,·)$, $\chi_{840}(671,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{23789766} a^{14} + \frac{724445}{7929922} a^{13} - \frac{20188}{99957} a^{12} - \frac{1044513}{7929922} a^{11} - \frac{626988}{3964961} a^{10} + \frac{982077}{3964961} a^{9} + \frac{1664972}{11894883} a^{8} - \frac{155489}{609994} a^{7} + \frac{1017179}{7929922} a^{6} + \frac{2915785}{7929922} a^{5} + \frac{4396663}{11894883} a^{4} + \frac{246010}{566423} a^{3} - \frac{3956989}{11894883} a^{2} + \frac{57947}{566423} a - \frac{6733}{27727}$, $\frac{1}{12201263055573746225272329382254} a^{15} + \frac{16003780233731838280445}{1355695895063749580585814375806} a^{14} + \frac{87757234586848241111195719663}{717721356210220366192489963662} a^{13} + \frac{8695063472039664341144234113}{2033543842595624370878721563709} a^{12} + \frac{446374932219454572232601052001}{2033543842595624370878721563709} a^{11} + \frac{142565924399622794325062805588}{677847947531874790292907187903} a^{10} - \frac{256457453482915109414296149775}{1109205732324886020479302671114} a^{9} + \frac{63926375507485637335338723535}{312852898860865287827495625186} a^{8} - \frac{174141801192854987074307898413}{4067087685191248741757443127418} a^{7} - \frac{880160677672769617266537929441}{4067087685191248741757443127418} a^{6} + \frac{995663050040593296656534516170}{6100631527786873112636164691127} a^{5} - \frac{916930346637894702341798901385}{4067087685191248741757443127418} a^{4} - \frac{28937604785974717201932359519}{554602866162443010239651335557} a^{3} - \frac{255518080691255028749850019009}{2033543842595624370878721563709} a^{2} + \frac{69119664909408097041375926486}{156426449430432643913747812593} a + \frac{48266142522729395189942423}{278835025722696334962117313}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{8}$, which has order $64$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{23312117946968840959975}{2380269811855978584719533629} a^{15} - \frac{431717345312550720390703}{17455311953610509621276579946} a^{14} + \frac{19121967307352111460880421}{52365935860831528863829739838} a^{13} - \frac{3338296112225062870847282}{2909218658935084936879429991} a^{12} + \frac{105537766304761498579742363}{17455311953610509621276579946} a^{11} - \frac{108540161510402356509459683}{5818437317870169873758859982} a^{10} + \frac{1241643006454052571896124956}{26182967930415764431914869919} a^{9} - \frac{405308877546574223623206483}{2909218658935084936879429991} a^{8} + \frac{131836313976680372559457481}{1026783056094735860075092938} a^{7} - \frac{4079603207347886872302874822}{8727655976805254810638289973} a^{6} - \frac{11983184904968195720400683059}{52365935860831528863829739838} a^{5} - \frac{2987225736731180144086758910}{2909218658935084936879429991} a^{4} - \frac{2945248416289971647771336584}{2014074456185828033224220763} a^{3} - \frac{8435657715617653867010822708}{2909218658935084936879429991} a^{2} - \frac{23384912391110308026722822395}{8727655976805254810638289973} a - \frac{85628478767291946194072293}{20344186426119475083072937} \) (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: | \( 288389.039928 \) (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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |