Normalized defining polynomial
\( x^{16} - 2 x^{15} - 18 x^{14} + 36 x^{13} + 272 x^{12} - 458 x^{11} - 1374 x^{10} - 736 x^{9} + 11490 x^{8} - 7916 x^{7} + 4680 x^{6} - 15094 x^{5} + 1565 x^{4} + 10890 x^{3} - 2640 x^{2} + 968 x + 1936 \)
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: | \(3167056956579818375390625=3^{8}\cdot 5^{8}\cdot 7^{8}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.99$ | 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: | $3, 5, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1155=3\cdot 5\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1155}(1,·)$, $\chi_{1155}(1154,·)$, $\chi_{1155}(769,·)$, $\chi_{1155}(76,·)$, $\chi_{1155}(461,·)$, $\chi_{1155}(386,·)$, $\chi_{1155}(274,·)$, $\chi_{1155}(659,·)$, $\chi_{1155}(736,·)$, $\chi_{1155}(1121,·)$, $\chi_{1155}(34,·)$, $\chi_{1155}(419,·)$, $\chi_{1155}(496,·)$, $\chi_{1155}(881,·)$, $\chi_{1155}(694,·)$, $\chi_{1155}(1079,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{336} a^{12} + \frac{1}{21} a^{11} + \frac{5}{84} a^{10} + \frac{17}{336} a^{9} + \frac{2}{21} a^{8} - \frac{1}{42} a^{7} + \frac{59}{336} a^{6} + \frac{2}{21} a^{5} - \frac{19}{84} a^{4} - \frac{11}{48} a^{3} - \frac{10}{21} a^{2} + \frac{5}{21} a - \frac{19}{42}$, $\frac{1}{1344} a^{13} + \frac{1}{1344} a^{12} - \frac{5}{672} a^{11} + \frac{53}{1344} a^{10} - \frac{13}{1344} a^{9} + \frac{25}{336} a^{8} - \frac{73}{1344} a^{7} + \frac{155}{1344} a^{6} + \frac{79}{672} a^{5} - \frac{197}{1344} a^{4} - \frac{559}{1344} a^{3} + \frac{29}{84} a^{2} + \frac{83}{168} a - \frac{17}{56}$, $\frac{1}{229965120} a^{14} - \frac{4693}{45993024} a^{13} + \frac{11567}{28745640} a^{12} + \frac{1861753}{229965120} a^{11} - \frac{2821061}{76655040} a^{10} - \frac{150223}{6763680} a^{9} - \frac{361173}{5110336} a^{8} + \frac{9895381}{229965120} a^{7} - \frac{690083}{6387920} a^{6} + \frac{53817647}{229965120} a^{5} + \frac{11631409}{76655040} a^{4} - \frac{3126373}{16426080} a^{3} - \frac{2697377}{28745640} a^{2} - \frac{257323}{522648} a + \frac{171331}{1306620}$, $\frac{1}{51392809687706625600} a^{15} - \frac{7710171587}{6424101210963328200} a^{14} - \frac{10219894502722909}{51392809687706625600} a^{13} - \frac{72924081336415543}{51392809687706625600} a^{12} + \frac{85483997836796161}{1511553226109018400} a^{11} + \frac{3135722450264770391}{51392809687706625600} a^{10} + \frac{3047234045031171617}{51392809687706625600} a^{9} - \frac{879511947421075151}{12848202421926656400} a^{8} - \frac{6438671662983248459}{51392809687706625600} a^{7} - \frac{2276112250192093457}{10278561937541325120} a^{6} + \frac{970016689080091049}{5139280968770662560} a^{5} - \frac{6374718258916033}{2234469986422027200} a^{4} - \frac{2616193088131594949}{8565468281284437600} a^{3} - \frac{15704037189318659}{62981384421209100} a^{2} - \frac{72835118815841951}{194669733665555400} a + \frac{10509644251269559}{292004600498333100}$
Class group and class number
$C_{2}\times C_{12}$, which has order $24$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{177963476429}{196663183204400} a^{15} - \frac{1344299276377}{589989549613200} a^{14} - \frac{35740271569657}{2359958198452800} a^{13} + \frac{95171695846811}{2359958198452800} a^{12} + \frac{266781406081951}{1179979099226400} a^{11} - \frac{416695822291119}{786652732817600} a^{10} - \frac{2317702591415159}{2359958198452800} a^{9} - \frac{9270851010779}{49165795801100} a^{8} + \frac{24968149675891793}{2359958198452800} a^{7} - \frac{1944471611832457}{157330546563520} a^{6} + \frac{2492461972104187}{235995819845280} a^{5} - \frac{699028316976503}{34202292731200} a^{4} + \frac{27858376853318663}{2359958198452800} a^{3} + \frac{259903076726459}{73748693701650} a^{2} - \frac{77111450405569}{26817706800600} a + \frac{88862163894889}{26817706800600} \) (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: | \( 442809.445079 \) (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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ | R | R | R | R | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 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 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |