Normalized defining polynomial
\( x^{42} - x^{41} - 61 x^{40} + 56 x^{39} + 1654 x^{38} - 1379 x^{37} - 26380 x^{36} + 19735 x^{35} + \cdots - 1 \)
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: | \(104\!\cdots\!125\) \(\medspace = 5^{21}\cdot 43^{40}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(80.38\) | 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: | $5^{1/2}43^{20/21}\approx 80.38393192209213$ | ||
Ramified primes: | \(5\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(215=5\cdot 43\) | ||
Dirichlet character group: | $\lbrace$$\chi_{215}(1,·)$, $\chi_{215}(4,·)$, $\chi_{215}(6,·)$, $\chi_{215}(9,·)$, $\chi_{215}(11,·)$, $\chi_{215}(14,·)$, $\chi_{215}(16,·)$, $\chi_{215}(146,·)$, $\chi_{215}(21,·)$, $\chi_{215}(24,·)$, $\chi_{215}(154,·)$, $\chi_{215}(31,·)$, $\chi_{215}(36,·)$, $\chi_{215}(41,·)$, $\chi_{215}(44,·)$, $\chi_{215}(176,·)$, $\chi_{215}(49,·)$, $\chi_{215}(181,·)$, $\chi_{215}(54,·)$, $\chi_{215}(56,·)$, $\chi_{215}(186,·)$, $\chi_{215}(59,·)$, $\chi_{215}(189,·)$, $\chi_{215}(64,·)$, $\chi_{215}(66,·)$, $\chi_{215}(139,·)$, $\chi_{215}(196,·)$, $\chi_{215}(74,·)$, $\chi_{215}(79,·)$, $\chi_{215}(81,·)$, $\chi_{215}(84,·)$, $\chi_{215}(164,·)$, $\chi_{215}(96,·)$, $\chi_{215}(144,·)$, $\chi_{215}(99,·)$, $\chi_{215}(101,·)$, $\chi_{215}(109,·)$, $\chi_{215}(111,·)$, $\chi_{215}(169,·)$, $\chi_{215}(121,·)$, $\chi_{215}(124,·)$, $\chi_{215}(126,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{24\!\cdots\!29}a^{41}-\frac{39\!\cdots\!65}{24\!\cdots\!29}a^{40}-\frac{31\!\cdots\!93}{24\!\cdots\!29}a^{39}+\frac{63\!\cdots\!01}{24\!\cdots\!29}a^{38}+\frac{67\!\cdots\!65}{24\!\cdots\!29}a^{37}-\frac{25\!\cdots\!10}{24\!\cdots\!29}a^{36}-\frac{80\!\cdots\!17}{24\!\cdots\!29}a^{35}+\frac{31\!\cdots\!95}{24\!\cdots\!29}a^{34}-\frac{10\!\cdots\!89}{24\!\cdots\!29}a^{33}+\frac{92\!\cdots\!95}{24\!\cdots\!29}a^{32}-\frac{62\!\cdots\!19}{24\!\cdots\!29}a^{31}+\frac{11\!\cdots\!14}{24\!\cdots\!29}a^{30}-\frac{84\!\cdots\!99}{24\!\cdots\!29}a^{29}+\frac{82\!\cdots\!69}{24\!\cdots\!29}a^{28}-\frac{78\!\cdots\!98}{24\!\cdots\!29}a^{27}+\frac{46\!\cdots\!21}{24\!\cdots\!29}a^{26}+\frac{10\!\cdots\!21}{24\!\cdots\!29}a^{25}-\frac{10\!\cdots\!03}{24\!\cdots\!29}a^{24}+\frac{35\!\cdots\!89}{24\!\cdots\!29}a^{23}-\frac{84\!\cdots\!02}{24\!\cdots\!29}a^{22}+\frac{67\!\cdots\!97}{24\!\cdots\!29}a^{21}-\frac{10\!\cdots\!00}{24\!\cdots\!29}a^{20}+\frac{34\!\cdots\!80}{24\!\cdots\!29}a^{19}-\frac{73\!\cdots\!17}{24\!\cdots\!29}a^{18}-\frac{75\!\cdots\!85}{24\!\cdots\!29}a^{17}+\frac{43\!\cdots\!94}{24\!\cdots\!29}a^{16}+\frac{33\!\cdots\!25}{24\!\cdots\!29}a^{15}-\frac{10\!\cdots\!42}{24\!\cdots\!29}a^{14}-\frac{17\!\cdots\!96}{24\!\cdots\!29}a^{13}+\frac{20\!\cdots\!19}{24\!\cdots\!29}a^{12}+\frac{53\!\cdots\!98}{24\!\cdots\!29}a^{11}+\frac{82\!\cdots\!42}{24\!\cdots\!29}a^{10}+\frac{24\!\cdots\!87}{24\!\cdots\!29}a^{9}+\frac{64\!\cdots\!59}{24\!\cdots\!29}a^{8}+\frac{18\!\cdots\!03}{24\!\cdots\!29}a^{7}-\frac{75\!\cdots\!39}{24\!\cdots\!29}a^{6}+\frac{46\!\cdots\!57}{24\!\cdots\!29}a^{5}-\frac{57\!\cdots\!99}{24\!\cdots\!29}a^{4}-\frac{11\!\cdots\!69}{24\!\cdots\!29}a^{3}+\frac{64\!\cdots\!28}{24\!\cdots\!29}a^{2}+\frac{92\!\cdots\!57}{24\!\cdots\!29}a+\frac{62\!\cdots\!41}{24\!\cdots\!29}$
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 $C_{42}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 3.3.1849.1, 6.6.427350125.1, 7.7.6321363049.1, 14.14.3121846156036138781328125.1, \(\Q(\zeta_{43})^+\) |
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 | ${\href{/padicField/2.14.0.1}{14} }^{3}$ | $42$ | R | ${\href{/padicField/7.6.0.1}{6} }^{7}$ | ${\href{/padicField/11.7.0.1}{7} }^{6}$ | $42$ | $42$ | $21^{2}$ | $42$ | $21^{2}$ | $21^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }^{6}$ | R | ${\href{/padicField/47.14.0.1}{14} }^{3}$ | $42$ | ${\href{/padicField/59.7.0.1}{7} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $42$ | $2$ | $21$ | $21$ | |||
\(43\) | Deg $42$ | $21$ | $2$ | $40$ |