Normalized defining polynomial
\( x^{42} - 16 x^{39} + 1066 x^{36} + 13384 x^{33} + 639480 x^{30} + 581064 x^{27} + 10535858 x^{24} + \cdots + 1 \)
Invariants
Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-506\!\cdots\!267\) \(\medspace = -\,3^{63}\cdot 29^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(93.15\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{3/2}29^{6/7}\approx 93.14635832670349$ | ||
Ramified primes: | \(3\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(261=3^{2}\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{261}(256,·)$, $\chi_{261}(1,·)$, $\chi_{261}(7,·)$, $\chi_{261}(136,·)$, $\chi_{261}(139,·)$, $\chi_{261}(140,·)$, $\chi_{261}(16,·)$, $\chi_{261}(146,·)$, $\chi_{261}(20,·)$, $\chi_{261}(23,·)$, $\chi_{261}(152,·)$, $\chi_{261}(25,·)$, $\chi_{261}(161,·)$, $\chi_{261}(169,·)$, $\chi_{261}(170,·)$, $\chi_{261}(257,·)$, $\chi_{261}(175,·)$, $\chi_{261}(49,·)$, $\chi_{261}(52,·)$, $\chi_{261}(53,·)$, $\chi_{261}(59,·)$, $\chi_{261}(190,·)$, $\chi_{261}(181,·)$, $\chi_{261}(65,·)$, $\chi_{261}(194,·)$, $\chi_{261}(197,·)$, $\chi_{261}(199,·)$, $\chi_{261}(74,·)$, $\chi_{261}(82,·)$, $\chi_{261}(83,·)$, $\chi_{261}(88,·)$, $\chi_{261}(94,·)$, $\chi_{261}(223,·)$, $\chi_{261}(226,·)$, $\chi_{261}(227,·)$, $\chi_{261}(103,·)$, $\chi_{261}(233,·)$, $\chi_{261}(107,·)$, $\chi_{261}(110,·)$, $\chi_{261}(239,·)$, $\chi_{261}(112,·)$, $\chi_{261}(248,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{1048576}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{171721640178611}a^{36}-\frac{83285078158257}{171721640178611}a^{33}-\frac{41211870895966}{171721640178611}a^{30}-\frac{55415557325085}{171721640178611}a^{27}+\frac{67310111522949}{171721640178611}a^{24}-\frac{34080177541187}{171721640178611}a^{21}+\frac{48573519142526}{171721640178611}a^{18}+\frac{77963249104841}{171721640178611}a^{15}-\frac{48617152457367}{171721640178611}a^{12}-\frac{22823869911138}{171721640178611}a^{9}+\frac{34054536605198}{171721640178611}a^{6}-\frac{54873007010306}{171721640178611}a^{3}+\frac{17212569281587}{171721640178611}$, $\frac{1}{171721640178611}a^{37}-\frac{83285078158257}{171721640178611}a^{34}-\frac{41211870895966}{171721640178611}a^{31}-\frac{55415557325085}{171721640178611}a^{28}+\frac{67310111522949}{171721640178611}a^{25}-\frac{34080177541187}{171721640178611}a^{22}+\frac{48573519142526}{171721640178611}a^{19}+\frac{77963249104841}{171721640178611}a^{16}-\frac{48617152457367}{171721640178611}a^{13}-\frac{22823869911138}{171721640178611}a^{10}+\frac{34054536605198}{171721640178611}a^{7}-\frac{54873007010306}{171721640178611}a^{4}+\frac{17212569281587}{171721640178611}a$, $\frac{1}{171721640178611}a^{38}-\frac{83285078158257}{171721640178611}a^{35}-\frac{41211870895966}{171721640178611}a^{32}-\frac{55415557325085}{171721640178611}a^{29}+\frac{67310111522949}{171721640178611}a^{26}-\frac{34080177541187}{171721640178611}a^{23}+\frac{48573519142526}{171721640178611}a^{20}+\frac{77963249104841}{171721640178611}a^{17}-\frac{48617152457367}{171721640178611}a^{14}-\frac{22823869911138}{171721640178611}a^{11}+\frac{34054536605198}{171721640178611}a^{8}-\frac{54873007010306}{171721640178611}a^{5}+\frac{17212569281587}{171721640178611}a^{2}$, $\frac{1}{26\!\cdots\!97}a^{39}-\frac{27\!\cdots\!95}{26\!\cdots\!97}a^{36}+\frac{69\!\cdots\!33}{26\!\cdots\!97}a^{33}+\frac{54\!\cdots\!48}{26\!\cdots\!97}a^{30}+\frac{59\!\cdots\!64}{26\!\cdots\!97}a^{27}+\frac{66\!\cdots\!18}{26\!\cdots\!97}a^{24}+\frac{17\!\cdots\!21}{26\!\cdots\!97}a^{21}+\frac{19\!\cdots\!57}{26\!\cdots\!97}a^{18}-\frac{10\!\cdots\!12}{26\!\cdots\!97}a^{15}+\frac{64\!\cdots\!56}{26\!\cdots\!97}a^{12}+\frac{26\!\cdots\!34}{15\!\cdots\!41}a^{9}-\frac{43\!\cdots\!22}{26\!\cdots\!97}a^{6}+\frac{12\!\cdots\!45}{26\!\cdots\!97}a^{3}+\frac{15\!\cdots\!03}{26\!\cdots\!97}$, $\frac{1}{26\!\cdots\!97}a^{40}-\frac{27\!\cdots\!95}{26\!\cdots\!97}a^{37}+\frac{69\!\cdots\!33}{26\!\cdots\!97}a^{34}+\frac{54\!\cdots\!48}{26\!\cdots\!97}a^{31}+\frac{59\!\cdots\!64}{26\!\cdots\!97}a^{28}+\frac{66\!\cdots\!18}{26\!\cdots\!97}a^{25}+\frac{17\!\cdots\!21}{26\!\cdots\!97}a^{22}+\frac{19\!\cdots\!57}{26\!\cdots\!97}a^{19}-\frac{10\!\cdots\!12}{26\!\cdots\!97}a^{16}+\frac{64\!\cdots\!56}{26\!\cdots\!97}a^{13}+\frac{26\!\cdots\!34}{15\!\cdots\!41}a^{10}-\frac{43\!\cdots\!22}{26\!\cdots\!97}a^{7}+\frac{12\!\cdots\!45}{26\!\cdots\!97}a^{4}+\frac{15\!\cdots\!03}{26\!\cdots\!97}a$, $\frac{1}{26\!\cdots\!97}a^{41}-\frac{27\!\cdots\!95}{26\!\cdots\!97}a^{38}+\frac{69\!\cdots\!33}{26\!\cdots\!97}a^{35}+\frac{54\!\cdots\!48}{26\!\cdots\!97}a^{32}+\frac{59\!\cdots\!64}{26\!\cdots\!97}a^{29}+\frac{66\!\cdots\!18}{26\!\cdots\!97}a^{26}+\frac{17\!\cdots\!21}{26\!\cdots\!97}a^{23}+\frac{19\!\cdots\!57}{26\!\cdots\!97}a^{20}-\frac{10\!\cdots\!12}{26\!\cdots\!97}a^{17}+\frac{64\!\cdots\!56}{26\!\cdots\!97}a^{14}+\frac{26\!\cdots\!34}{15\!\cdots\!41}a^{11}-\frac{43\!\cdots\!22}{26\!\cdots\!97}a^{8}+\frac{12\!\cdots\!45}{26\!\cdots\!97}a^{5}+\frac{15\!\cdots\!03}{26\!\cdots\!97}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{506555599156652661449309472991747718983058900032}{157418057130855963202750560875684372674171552118941} a^{40} - \frac{8114583616242293050940009467933134571260805848818}{157418057130855963202750560875684372674171552118941} a^{37} + \frac{540143405824382084789333222256098190206354875785817}{157418057130855963202750560875684372674171552118941} a^{34} + \frac{6769405777152002546262368018472930814863807568729202}{157418057130855963202750560875684372674171552118941} a^{31} + \frac{323802464519563768341463442999828448581845183651136849}{157418057130855963202750560875684372674171552118941} a^{28} + \frac{288142517330108029904735240247623985318031699094195776}{157418057130855963202750560875684372674171552118941} a^{25} + \frac{5331385835500769726808039233660652606344991507052105504}{157418057130855963202750560875684372674171552118941} a^{22} - \frac{12969798271891141628500682868998519653854136096029691491}{157418057130855963202750560875684372674171552118941} a^{19} + \frac{87275844225049443301551336843866278720782741443613971589}{157418057130855963202750560875684372674171552118941} a^{16} - \frac{95211707577760126850777186038687512368139969173505476623}{157418057130855963202750560875684372674171552118941} a^{13} + \frac{102190137672910470361389908578234127631442163567919249487}{157418057130855963202750560875684372674171552118941} a^{10} - \frac{27539214918725399502213103022121123741017756427717838}{157418057130855963202750560875684372674171552118941} a^{7} - \frac{128239545942073126390259283430967160595130463488664}{157418057130855963202750560875684372674171552118941} a^{4} - \frac{927575793299550729533637408652827313433016575552919}{157418057130855963202750560875684372674171552118941} a \) (order $18$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
A cyclic group of order 42 |
The 42 conjugacy class representatives for $C_{42}$ |
Character table for $C_{42}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 7.7.594823321.1, 14.0.773792930870360792667.1, 21.21.4814587615056751193058435502319478353721.1 |
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 | $42$ | R | $42$ | $21^{2}$ | $42$ | $21^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{21}$ | ${\href{/padicField/19.7.0.1}{7} }^{6}$ | $42$ | R | $21^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{7}$ | $21^{2}$ | $42$ | ${\href{/padicField/53.14.0.1}{14} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $42$ | $6$ | $7$ | $63$ | |||
\(29\) | Deg $42$ | $7$ | $6$ | $36$ |