Normalized defining polynomial
\( x^{42} - 21 x^{41} + 231 x^{40} - 1750 x^{39} + 10395 x^{38} - 52269 x^{37} + 233611 x^{36} + \cdots + 99519182315771 \)
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: | \(-124\!\cdots\!811\) \(\medspace = -\,7^{76}\cdot 11^{21}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(112.18\) | 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: | $7^{38/21}11^{1/2}\approx 112.18153219054884$ | ||
Ramified primes: | \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(539=7^{2}\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{539}(1,·)$, $\chi_{539}(386,·)$, $\chi_{539}(263,·)$, $\chi_{539}(142,·)$, $\chi_{539}(527,·)$, $\chi_{539}(144,·)$, $\chi_{539}(529,·)$, $\chi_{539}(274,·)$, $\chi_{539}(23,·)$, $\chi_{539}(408,·)$, $\chi_{539}(155,·)$, $\chi_{539}(32,·)$, $\chi_{539}(417,·)$, $\chi_{539}(296,·)$, $\chi_{539}(298,·)$, $\chi_{539}(43,·)$, $\chi_{539}(428,·)$, $\chi_{539}(177,·)$, $\chi_{539}(309,·)$, $\chi_{539}(186,·)$, $\chi_{539}(65,·)$, $\chi_{539}(450,·)$, $\chi_{539}(67,·)$, $\chi_{539}(452,·)$, $\chi_{539}(197,·)$, $\chi_{539}(331,·)$, $\chi_{539}(78,·)$, $\chi_{539}(463,·)$, $\chi_{539}(340,·)$, $\chi_{539}(219,·)$, $\chi_{539}(221,·)$, $\chi_{539}(351,·)$, $\chi_{539}(100,·)$, $\chi_{539}(485,·)$, $\chi_{539}(232,·)$, $\chi_{539}(109,·)$, $\chi_{539}(494,·)$, $\chi_{539}(373,·)$, $\chi_{539}(375,·)$, $\chi_{539}(120,·)$, $\chi_{539}(505,·)$, $\chi_{539}(254,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{1048576}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $\frac{1}{97}a^{39}-\frac{29}{97}a^{38}-\frac{37}{97}a^{37}-\frac{8}{97}a^{36}+\frac{28}{97}a^{35}+\frac{42}{97}a^{34}+\frac{18}{97}a^{33}-\frac{4}{97}a^{32}+\frac{47}{97}a^{31}+\frac{34}{97}a^{30}-\frac{16}{97}a^{29}-\frac{36}{97}a^{28}+\frac{30}{97}a^{27}-\frac{16}{97}a^{26}-\frac{35}{97}a^{25}-\frac{43}{97}a^{24}-\frac{47}{97}a^{23}+\frac{12}{97}a^{22}-\frac{6}{97}a^{21}+\frac{8}{97}a^{20}+\frac{23}{97}a^{19}+\frac{17}{97}a^{18}-\frac{20}{97}a^{17}-\frac{47}{97}a^{16}+\frac{16}{97}a^{15}-\frac{25}{97}a^{14}-\frac{6}{97}a^{13}+\frac{41}{97}a^{12}+\frac{14}{97}a^{11}+\frac{26}{97}a^{10}-\frac{25}{97}a^{9}-\frac{21}{97}a^{8}+\frac{11}{97}a^{7}+\frac{11}{97}a^{6}-\frac{6}{97}a^{5}-\frac{6}{97}a^{4}-\frac{26}{97}a^{3}-\frac{46}{97}a^{2}+\frac{36}{97}a+\frac{45}{97}$, $\frac{1}{97}a^{40}-\frac{5}{97}a^{38}-\frac{14}{97}a^{37}-\frac{10}{97}a^{36}-\frac{19}{97}a^{35}-\frac{25}{97}a^{34}+\frac{33}{97}a^{33}+\frac{28}{97}a^{32}+\frac{39}{97}a^{31}-\frac{15}{97}a^{29}-\frac{44}{97}a^{28}-\frac{19}{97}a^{27}-\frac{14}{97}a^{26}+\frac{9}{97}a^{25}-\frac{33}{97}a^{24}+\frac{7}{97}a^{23}-\frac{46}{97}a^{22}+\frac{28}{97}a^{21}-\frac{36}{97}a^{20}+\frac{5}{97}a^{19}-\frac{12}{97}a^{18}-\frac{45}{97}a^{17}+\frac{11}{97}a^{16}-\frac{46}{97}a^{15}+\frac{45}{97}a^{14}-\frac{36}{97}a^{13}+\frac{39}{97}a^{12}+\frac{44}{97}a^{11}-\frac{47}{97}a^{10}+\frac{30}{97}a^{9}-\frac{16}{97}a^{8}+\frac{39}{97}a^{7}+\frac{22}{97}a^{6}+\frac{14}{97}a^{5}-\frac{6}{97}a^{4}-\frac{24}{97}a^{3}-\frac{37}{97}a^{2}+\frac{22}{97}a+\frac{44}{97}$, $\frac{1}{47\!\cdots\!67}a^{41}-\frac{31\!\cdots\!60}{47\!\cdots\!67}a^{40}-\frac{19\!\cdots\!10}{47\!\cdots\!67}a^{39}-\frac{11\!\cdots\!71}{47\!\cdots\!67}a^{38}+\frac{23\!\cdots\!21}{47\!\cdots\!67}a^{37}-\frac{22\!\cdots\!22}{47\!\cdots\!67}a^{36}-\frac{22\!\cdots\!29}{47\!\cdots\!67}a^{35}+\frac{26\!\cdots\!94}{47\!\cdots\!67}a^{34}-\frac{11\!\cdots\!34}{47\!\cdots\!67}a^{33}-\frac{70\!\cdots\!76}{47\!\cdots\!67}a^{32}+\frac{99\!\cdots\!29}{47\!\cdots\!67}a^{31}-\frac{70\!\cdots\!31}{47\!\cdots\!67}a^{30}-\frac{18\!\cdots\!93}{47\!\cdots\!67}a^{29}+\frac{38\!\cdots\!96}{47\!\cdots\!67}a^{28}+\frac{43\!\cdots\!34}{47\!\cdots\!67}a^{27}+\frac{45\!\cdots\!64}{47\!\cdots\!67}a^{26}+\frac{20\!\cdots\!24}{47\!\cdots\!67}a^{25}-\frac{38\!\cdots\!57}{47\!\cdots\!67}a^{24}+\frac{11\!\cdots\!78}{47\!\cdots\!67}a^{23}-\frac{34\!\cdots\!62}{47\!\cdots\!67}a^{22}+\frac{20\!\cdots\!52}{47\!\cdots\!67}a^{21}+\frac{56\!\cdots\!93}{47\!\cdots\!67}a^{20}-\frac{75\!\cdots\!70}{47\!\cdots\!67}a^{19}-\frac{14\!\cdots\!94}{47\!\cdots\!67}a^{18}+\frac{14\!\cdots\!83}{47\!\cdots\!67}a^{17}+\frac{95\!\cdots\!37}{47\!\cdots\!67}a^{16}+\frac{10\!\cdots\!97}{47\!\cdots\!67}a^{15}+\frac{31\!\cdots\!97}{47\!\cdots\!67}a^{14}+\frac{60\!\cdots\!40}{47\!\cdots\!67}a^{13}+\frac{86\!\cdots\!58}{47\!\cdots\!67}a^{12}+\frac{53\!\cdots\!02}{47\!\cdots\!67}a^{11}-\frac{96\!\cdots\!28}{47\!\cdots\!67}a^{10}-\frac{24\!\cdots\!67}{47\!\cdots\!67}a^{9}-\frac{78\!\cdots\!71}{47\!\cdots\!67}a^{8}-\frac{16\!\cdots\!36}{47\!\cdots\!67}a^{7}+\frac{23\!\cdots\!16}{47\!\cdots\!67}a^{6}+\frac{48\!\cdots\!16}{47\!\cdots\!67}a^{5}+\frac{17\!\cdots\!11}{47\!\cdots\!67}a^{4}-\frac{19\!\cdots\!22}{47\!\cdots\!67}a^{3}+\frac{10\!\cdots\!25}{47\!\cdots\!67}a^{2}-\frac{11\!\cdots\!46}{47\!\cdots\!67}a-\frac{20\!\cdots\!34}{47\!\cdots\!67}$
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: | \( -1 \) (order $2$) | 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 $C_{42}$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{7})^+\), 6.0.3195731.1, 7.7.13841287201.1, 14.0.3733376216303663794289149571.1, \(\Q(\zeta_{49})^+\) |
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 | $42$ | $21^{2}$ | $21^{2}$ | R | R | ${\href{/padicField/13.14.0.1}{14} }^{3}$ | $42$ | ${\href{/padicField/19.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{14}$ | $21^{2}$ | ${\href{/padicField/41.14.0.1}{14} }^{3}$ | ${\href{/padicField/43.14.0.1}{14} }^{3}$ | $21^{2}$ | $21^{2}$ | $21^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | Deg $42$ | $21$ | $2$ | $76$ | |||
\(11\) | Deg $42$ | $2$ | $21$ | $21$ |