Normalized defining polynomial
\( x^{42} - 105 x^{40} + 4977 x^{38} - 140819 x^{36} - 2 x^{35} + 2650095 x^{34} + 28 x^{33} - 34991649 x^{32} + 3416 x^{31} + 333300632 x^{30} - 147910 x^{29} - 2319579309 x^{28} + \cdots + 1 \)
Invariants
Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[42, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(776\!\cdots\!312\) \(\medspace = 2^{42}\cdot 3^{21}\cdot 7^{76}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(117.17\) | 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: | $2\cdot 3^{1/2}7^{38/21}\approx 117.16978899149642$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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: | \(588=2^{2}\cdot 3\cdot 7^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{588}(1,·)$, $\chi_{588}(515,·)$, $\chi_{588}(263,·)$, $\chi_{588}(11,·)$, $\chi_{588}(527,·)$, $\chi_{588}(529,·)$, $\chi_{588}(275,·)$, $\chi_{588}(277,·)$, $\chi_{588}(23,·)$, $\chi_{588}(25,·)$, $\chi_{588}(155,·)$, $\chi_{588}(541,·)$, $\chi_{588}(289,·)$, $\chi_{588}(421,·)$, $\chi_{588}(169,·)$, $\chi_{588}(95,·)$, $\chi_{588}(431,·)$, $\chi_{588}(179,·)$, $\chi_{588}(443,·)$, $\chi_{588}(445,·)$, $\chi_{588}(575,·)$, $\chi_{588}(193,·)$, $\chi_{588}(323,·)$, $\chi_{588}(407,·)$, $\chi_{588}(71,·)$, $\chi_{588}(457,·)$, $\chi_{588}(205,·)$, $\chi_{588}(337,·)$, $\chi_{588}(85,·)$, $\chi_{588}(121,·)$, $\chi_{588}(347,·)$, $\chi_{588}(37,·)$, $\chi_{588}(107,·)$, $\chi_{588}(359,·)$, $\chi_{588}(361,·)$, $\chi_{588}(491,·)$, $\chi_{588}(109,·)$, $\chi_{588}(239,·)$, $\chi_{588}(373,·)$, $\chi_{588}(505,·)$, $\chi_{588}(191,·)$, $\chi_{588}(253,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{97}a^{37}+\frac{45}{97}a^{36}+\frac{16}{97}a^{35}+\frac{34}{97}a^{34}+\frac{18}{97}a^{33}-\frac{26}{97}a^{32}+\frac{45}{97}a^{31}-\frac{27}{97}a^{30}-\frac{24}{97}a^{29}+\frac{22}{97}a^{28}-\frac{6}{97}a^{27}-\frac{19}{97}a^{26}+\frac{20}{97}a^{25}+\frac{35}{97}a^{24}-\frac{37}{97}a^{23}+\frac{39}{97}a^{22}+\frac{46}{97}a^{21}-\frac{20}{97}a^{20}+\frac{12}{97}a^{19}-\frac{27}{97}a^{18}+\frac{39}{97}a^{17}+\frac{2}{97}a^{16}+\frac{18}{97}a^{15}-\frac{37}{97}a^{14}+\frac{33}{97}a^{13}-\frac{6}{97}a^{12}+\frac{22}{97}a^{11}+\frac{12}{97}a^{10}-\frac{34}{97}a^{9}-\frac{31}{97}a^{8}-\frac{46}{97}a^{7}+\frac{28}{97}a^{6}-\frac{24}{97}a^{5}-\frac{18}{97}a^{4}-\frac{48}{97}a^{3}+\frac{28}{97}a^{2}+\frac{11}{97}a+\frac{1}{97}$, $\frac{1}{97}a^{38}+\frac{28}{97}a^{36}-\frac{7}{97}a^{35}+\frac{40}{97}a^{34}+\frac{37}{97}a^{33}-\frac{46}{97}a^{32}-\frac{15}{97}a^{31}+\frac{27}{97}a^{30}+\frac{35}{97}a^{29}-\frac{26}{97}a^{28}-\frac{40}{97}a^{27}+\frac{2}{97}a^{26}+\frac{8}{97}a^{25}+\frac{37}{97}a^{24}-\frac{42}{97}a^{23}+\frac{37}{97}a^{22}+\frac{44}{97}a^{21}+\frac{39}{97}a^{20}+\frac{15}{97}a^{19}-\frac{7}{97}a^{18}-\frac{7}{97}a^{17}+\frac{25}{97}a^{16}+\frac{26}{97}a^{15}-\frac{48}{97}a^{14}-\frac{36}{97}a^{13}+\frac{1}{97}a^{12}-\frac{8}{97}a^{11}+\frac{8}{97}a^{10}+\frac{44}{97}a^{9}-\frac{9}{97}a^{8}-\frac{36}{97}a^{7}-\frac{23}{97}a^{6}-\frac{5}{97}a^{5}-\frac{14}{97}a^{4}-\frac{43}{97}a^{3}+\frac{12}{97}a^{2}-\frac{9}{97}a-\frac{45}{97}$, $\frac{1}{97}a^{39}-\frac{6}{97}a^{36}-\frac{20}{97}a^{35}-\frac{42}{97}a^{34}+\frac{32}{97}a^{33}+\frac{34}{97}a^{32}+\frac{28}{97}a^{31}+\frac{15}{97}a^{30}-\frac{33}{97}a^{29}+\frac{23}{97}a^{28}-\frac{24}{97}a^{27}-\frac{42}{97}a^{26}-\frac{38}{97}a^{25}+\frac{45}{97}a^{24}+\frac{6}{97}a^{23}+\frac{19}{97}a^{22}+\frac{12}{97}a^{21}-\frac{7}{97}a^{20}+\frac{45}{97}a^{19}-\frac{27}{97}a^{18}-\frac{30}{97}a^{16}+\frac{30}{97}a^{15}+\frac{30}{97}a^{14}+\frac{47}{97}a^{13}-\frac{34}{97}a^{12}-\frac{26}{97}a^{11}-\frac{1}{97}a^{10}-\frac{27}{97}a^{9}-\frac{41}{97}a^{8}+\frac{4}{97}a^{7}-\frac{13}{97}a^{6}-\frac{21}{97}a^{5}-\frac{24}{97}a^{4}-\frac{2}{97}a^{3}-\frac{17}{97}a^{2}+\frac{35}{97}a-\frac{28}{97}$, $\frac{1}{97}a^{40}-\frac{41}{97}a^{36}-\frac{43}{97}a^{35}+\frac{42}{97}a^{34}+\frac{45}{97}a^{33}-\frac{31}{97}a^{32}-\frac{6}{97}a^{31}-\frac{1}{97}a^{30}-\frac{24}{97}a^{29}+\frac{11}{97}a^{28}+\frac{19}{97}a^{27}+\frac{42}{97}a^{26}-\frac{29}{97}a^{25}+\frac{22}{97}a^{24}-\frac{9}{97}a^{23}-\frac{45}{97}a^{22}-\frac{22}{97}a^{21}+\frac{22}{97}a^{20}+\frac{45}{97}a^{19}+\frac{32}{97}a^{18}+\frac{10}{97}a^{17}+\frac{42}{97}a^{16}+\frac{41}{97}a^{15}+\frac{19}{97}a^{14}-\frac{30}{97}a^{13}+\frac{35}{97}a^{12}+\frac{34}{97}a^{11}+\frac{45}{97}a^{10}+\frac{46}{97}a^{9}+\frac{12}{97}a^{8}+\frac{2}{97}a^{7}-\frac{47}{97}a^{6}+\frac{26}{97}a^{5}-\frac{13}{97}a^{4}-\frac{14}{97}a^{3}+\frac{9}{97}a^{2}+\frac{38}{97}a+\frac{6}{97}$, $\frac{1}{21\!\cdots\!87}a^{41}+\frac{10\!\cdots\!59}{21\!\cdots\!87}a^{40}-\frac{10\!\cdots\!41}{21\!\cdots\!87}a^{39}+\frac{10\!\cdots\!29}{21\!\cdots\!87}a^{38}-\frac{10\!\cdots\!65}{21\!\cdots\!87}a^{37}+\frac{95\!\cdots\!03}{21\!\cdots\!87}a^{36}+\frac{28\!\cdots\!67}{21\!\cdots\!87}a^{35}+\frac{74\!\cdots\!98}{21\!\cdots\!87}a^{34}+\frac{69\!\cdots\!90}{21\!\cdots\!87}a^{33}+\frac{57\!\cdots\!14}{21\!\cdots\!87}a^{32}-\frac{59\!\cdots\!05}{21\!\cdots\!87}a^{31}-\frac{51\!\cdots\!36}{21\!\cdots\!87}a^{30}+\frac{20\!\cdots\!62}{21\!\cdots\!87}a^{29}-\frac{42\!\cdots\!19}{21\!\cdots\!87}a^{28}-\frac{69\!\cdots\!29}{21\!\cdots\!87}a^{27}-\frac{88\!\cdots\!22}{21\!\cdots\!87}a^{26}-\frac{30\!\cdots\!61}{21\!\cdots\!87}a^{25}+\frac{86\!\cdots\!40}{21\!\cdots\!87}a^{24}-\frac{48\!\cdots\!25}{21\!\cdots\!87}a^{23}+\frac{56\!\cdots\!42}{21\!\cdots\!87}a^{22}-\frac{92\!\cdots\!13}{21\!\cdots\!87}a^{21}-\frac{42\!\cdots\!38}{21\!\cdots\!87}a^{20}-\frac{94\!\cdots\!75}{21\!\cdots\!87}a^{19}-\frac{42\!\cdots\!20}{21\!\cdots\!87}a^{18}+\frac{28\!\cdots\!76}{21\!\cdots\!87}a^{17}-\frac{66\!\cdots\!98}{21\!\cdots\!87}a^{16}-\frac{51\!\cdots\!79}{21\!\cdots\!87}a^{15}+\frac{39\!\cdots\!93}{21\!\cdots\!87}a^{14}+\frac{77\!\cdots\!98}{21\!\cdots\!87}a^{13}+\frac{61\!\cdots\!78}{21\!\cdots\!87}a^{12}-\frac{77\!\cdots\!21}{21\!\cdots\!87}a^{11}+\frac{17\!\cdots\!88}{21\!\cdots\!87}a^{10}-\frac{76\!\cdots\!70}{21\!\cdots\!87}a^{9}-\frac{11\!\cdots\!10}{21\!\cdots\!87}a^{8}+\frac{64\!\cdots\!15}{21\!\cdots\!87}a^{7}-\frac{67\!\cdots\!13}{21\!\cdots\!87}a^{6}+\frac{75\!\cdots\!76}{21\!\cdots\!87}a^{5}+\frac{47\!\cdots\!95}{21\!\cdots\!87}a^{4}+\frac{10\!\cdots\!73}{21\!\cdots\!87}a^{3}-\frac{65\!\cdots\!68}{21\!\cdots\!87}a^{2}+\frac{38\!\cdots\!55}{21\!\cdots\!87}a-\frac{28\!\cdots\!63}{21\!\cdots\!87}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $41$ | 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 $C_{42}$ is not computed |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\zeta_{7})^+\), 6.6.4148928.1, 7.7.13841287201.1, 14.14.6864701899232030692065067008.1, \(\Q(\zeta_{49})^+\) |
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 | R | R | $42$ | R | $21^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{6}$ | $42$ | ${\href{/padicField/19.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/41.14.0.1}{14} }^{3}$ | ${\href{/padicField/43.14.0.1}{14} }^{3}$ | $21^{2}$ | $42$ | $21^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $42$ | $2$ | $21$ | $42$ | |||
\(3\) | Deg $42$ | $2$ | $21$ | $21$ | |||
\(7\) | Deg $42$ | $21$ | $2$ | $76$ |