Normalized defining polynomial
\( x^{16} - 4 x^{15} - 10 x^{14} + 24 x^{13} + 261 x^{12} - 608 x^{11} - 466 x^{10} + 64 x^{9} + 6713 x^{8} - 9704 x^{7} + 6186 x^{6} - 11232 x^{5} + 15880 x^{4} - 10608 x^{3} + 6552 x^{2} - 4320 x + 1296 \)
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: | \(247875891108249600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.98$ | 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$$\chi_{840}(1,·)$, $\chi_{840}(491,·)$, $\chi_{840}(769,·)$, $\chi_{840}(139,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(601,·)$, $\chi_{840}(281,·)$, $\chi_{840}(419,·)$, $\chi_{840}(379,·)$, $\chi_{840}(169,·)$, $\chi_{840}(811,·)$, $\chi_{840}(449,·)$, $\chi_{840}(659,·)$, $\chi_{840}(41,·)$, $\chi_{840}(251,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{30} a^{11} + \frac{1}{30} a^{9} + \frac{1}{30} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{1}{3} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{13}{30} a^{2} - \frac{4}{15} a + \frac{1}{5}$, $\frac{1}{60} a^{12} - \frac{1}{30} a^{10} + \frac{1}{15} a^{9} - \frac{3}{20} a^{8} + \frac{1}{5} a^{7} - \frac{1}{15} a^{6} - \frac{1}{5} a^{5} + \frac{9}{20} a^{4} - \frac{2}{15} a^{3} - \frac{7}{30} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{180} a^{13} + \frac{1}{180} a^{12} - \frac{1}{90} a^{11} + \frac{1}{90} a^{10} - \frac{1}{36} a^{9} + \frac{11}{60} a^{8} + \frac{19}{90} a^{7} - \frac{4}{45} a^{6} - \frac{5}{12} a^{5} - \frac{41}{180} a^{4} + \frac{17}{45} a^{3} + \frac{16}{45} a^{2} + \frac{2}{5}$, $\frac{1}{39600} a^{14} + \frac{1}{1650} a^{13} - \frac{1}{3300} a^{12} + \frac{29}{4950} a^{11} + \frac{241}{7920} a^{10} + \frac{101}{9900} a^{9} + \frac{2399}{9900} a^{8} + \frac{82}{825} a^{7} + \frac{5017}{39600} a^{6} - \frac{3413}{9900} a^{5} + \frac{2099}{4950} a^{4} + \frac{2059}{4950} a^{3} - \frac{329}{4950} a^{2} - \frac{199}{825} a + \frac{127}{550}$, $\frac{1}{531821188521685537200} a^{15} - \frac{568717366245541}{132955297130421384300} a^{14} + \frac{15764129840064431}{33238824282605346075} a^{13} - \frac{34479099768519434}{11079608094201782025} a^{12} - \frac{2608547437298541}{371643038799221200} a^{11} - \frac{106547908285341401}{3910449915600628950} a^{10} + \frac{6471197025721976623}{66477648565210692150} a^{9} + \frac{1553712747171305681}{66477648565210692150} a^{8} - \frac{1811720802003287191}{531821188521685537200} a^{7} + \frac{4851783625313759819}{66477648565210692150} a^{6} + \frac{8013219057119290919}{44318432376807128100} a^{5} + \frac{730898150389285133}{7386405396134521350} a^{4} - \frac{19676007038911345951}{66477648565210692150} a^{3} - \frac{3920637006770310179}{22159216188403564050} a^{2} - \frac{123379548469361071}{1477281079226904270} a - \frac{287006496767277086}{1231067566022420225}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{24321981916923431}{9669476154939737040} a^{15} + \frac{60527686683138899}{9669476154939737040} a^{14} + \frac{86208439124073119}{2417369038734934260} a^{13} - \frac{1822932440161727}{201447419894577855} a^{12} - \frac{4628522923109141}{6757146159985840} a^{11} + \frac{282744406707628907}{568792714996455120} a^{10} + \frac{2667886003706996203}{1208684519367467130} a^{9} + \frac{1798528859339735488}{604342259683733565} a^{8} - \frac{128401126857343730143}{9669476154939737040} a^{7} + \frac{29451486416072973631}{9669476154939737040} a^{6} - \frac{4475301637703162797}{805789679578311420} a^{5} + \frac{997177238712518723}{53719311971887428} a^{4} - \frac{1081384809208451251}{120868451936746713} a^{3} + \frac{2097977440447360627}{402894839789155710} a^{2} - \frac{555136891671696971}{134298279929718570} a + \frac{64237729386469487}{44766093309906190} \) (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: | \( 254690.329412 \) (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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| $3$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |