Normalized defining polynomial
\( x^{16} + 6 x^{14} - 20 x^{13} + 119 x^{12} + 280 x^{11} + 104 x^{10} - 140 x^{9} - 6396 x^{8} - 480 x^{7} - 11434 x^{6} - 62260 x^{5} + 465079 x^{4} - 481080 x^{3} - 40716 x^{2} - 171900 x + 365481 \)
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$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(589,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(601,·)$, $\chi_{840}(29,·)$, $\chi_{840}(671,·)$, $\chi_{840}(419,·)$, $\chi_{840}(41,·)$, $\chi_{840}(71,·)$, $\chi_{840}(349,·)$, $\chi_{840}(629,·)$, $\chi_{840}(631,·)$, $\chi_{840}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4440} a^{12} + \frac{31}{111} a^{11} + \frac{1}{148} a^{10} + \frac{21}{74} a^{9} + \frac{143}{444} a^{8} + \frac{2}{37} a^{7} + \frac{239}{740} a^{6} + \frac{19}{111} a^{5} - \frac{17}{444} a^{4} - \frac{17}{37} a^{3} + \frac{47}{222} a^{2} - \frac{31}{74} a + \frac{643}{1480}$, $\frac{1}{4440} a^{13} - \frac{133}{444} a^{11} - \frac{7}{74} a^{10} + \frac{191}{444} a^{9} - \frac{35}{111} a^{8} + \frac{219}{740} a^{7} - \frac{35}{111} a^{6} - \frac{43}{148} a^{5} + \frac{2}{111} a^{4} - \frac{13}{222} a^{3} + \frac{13}{222} a^{2} - \frac{157}{1480} a + \frac{10}{37}$, $\frac{1}{1272321960} a^{14} - \frac{20701}{212053660} a^{13} + \frac{433}{254464392} a^{12} - \frac{1060315}{10602683} a^{11} + \frac{9426149}{31808049} a^{10} + \frac{15851305}{31808049} a^{9} + \frac{5693927}{106026830} a^{8} + \frac{55941049}{318080490} a^{7} + \frac{349677}{3029338} a^{6} + \frac{26030419}{63616098} a^{5} - \frac{3576155}{18176028} a^{4} + \frac{2366443}{9088014} a^{3} + \frac{2318423}{424107320} a^{2} + \frac{78809}{644540} a - \frac{35141935}{84821464}$, $\frac{1}{124912285448691547937296617360} a^{15} + \frac{986836571211787031}{124912285448691547937296617360} a^{14} - \frac{1959703052793832264395637}{17844612206955935419613802480} a^{13} + \frac{74827660225593390543763}{1125335904943167098534203760} a^{12} - \frac{234826719804825738638087131}{6245614272434577396864830868} a^{11} - \frac{391954985869854236991225332}{1561403568108644349216207717} a^{10} + \frac{837429767646476952916060399}{2230576525869491927451725310} a^{9} + \frac{55786281066949374943698079}{2230576525869491927451725310} a^{8} - \frac{10887161009922641831234616709}{31228071362172886984324154340} a^{7} + \frac{1496035989669727946754185402}{7807017840543221746081038585} a^{6} + \frac{5307306359750418293005510687}{12491228544869154793729661736} a^{5} + \frac{20253682732957339187131633}{45755415915271629281061032} a^{4} - \frac{1555265059883080819267270763}{3376007714829501295602611280} a^{3} + \frac{47657801249344096497280965239}{124912285448691547937296617360} a^{2} - \frac{19593521449699954503685742007}{41637428482897182645765539120} a + \frac{17021625326570121984172235749}{41637428482897182645765539120}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1467357934192016651}{67286905149172920000720} a^{15} + \frac{651745410777945}{18518678245776393776} a^{14} + \frac{23162236854348527987}{124961395277035422858480} a^{13} - \frac{3193086318467588281}{23641345052412107027280} a^{12} + \frac{25795501061315699926}{10934122086740599500117} a^{11} + \frac{435570573732561990955}{43736488346962398000468} a^{10} + \frac{562440962005168190339}{31240348819258855714620} a^{9} + \frac{160138381319918199059}{6248069763851771142924} a^{8} - \frac{3587229231676385922161}{36447073622468665000390} a^{7} - \frac{37280335080477611752927}{218682441734811990002340} a^{6} - \frac{45157573908981549402761}{87472976693924796000936} a^{5} - \frac{27546666565331648514991}{12496139527703542285848} a^{4} + \frac{156653802679016612542171}{23641345052412107027280} a^{3} + \frac{67370715274015358573351}{174945953387849592001872} a^{2} - \frac{348998779758906887896803}{291576588979749320003120} a - \frac{1242442843562157995682681}{291576588979749320003120} \) (order $12$) | 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: | \( 829665.224905 \) (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 | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |