Normalized defining polynomial
\( x^{40} + 243x^{30} + 59049x^{20} + 14348907x^{10} + 3486784401 \)
Invariants
Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(32473210254684090614318847656250000000000000000000000000000000000000000\) \(\medspace = 2^{40}\cdot 3^{20}\cdot 5^{70}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(57.91\) | 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: | $2\cdot 3^{1/2}5^{7/4}\approx 57.91460926441345$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(300=2^{2}\cdot 3\cdot 5^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(131,·)$, $\chi_{300}(133,·)$, $\chi_{300}(263,·)$, $\chi_{300}(11,·)$, $\chi_{300}(13,·)$, $\chi_{300}(143,·)$, $\chi_{300}(277,·)$, $\chi_{300}(23,·)$, $\chi_{300}(157,·)$, $\chi_{300}(287,·)$, $\chi_{300}(289,·)$, $\chi_{300}(37,·)$, $\chi_{300}(167,·)$, $\chi_{300}(169,·)$, $\chi_{300}(299,·)$, $\chi_{300}(47,·)$, $\chi_{300}(49,·)$, $\chi_{300}(179,·)$, $\chi_{300}(181,·)$, $\chi_{300}(59,·)$, $\chi_{300}(61,·)$, $\chi_{300}(191,·)$, $\chi_{300}(193,·)$, $\chi_{300}(71,·)$, $\chi_{300}(73,·)$, $\chi_{300}(203,·)$, $\chi_{300}(83,·)$, $\chi_{300}(217,·)$, $\chi_{300}(97,·)$, $\chi_{300}(227,·)$, $\chi_{300}(229,·)$, $\chi_{300}(107,·)$, $\chi_{300}(109,·)$, $\chi_{300}(239,·)$, $\chi_{300}(241,·)$, $\chi_{300}(119,·)$, $\chi_{300}(121,·)$, $\chi_{300}(251,·)$, $\chi_{300}(253,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$, $\frac{1}{59049}a^{20}$, $\frac{1}{59049}a^{21}$, $\frac{1}{177147}a^{22}$, $\frac{1}{177147}a^{23}$, $\frac{1}{531441}a^{24}$, $\frac{1}{531441}a^{25}$, $\frac{1}{1594323}a^{26}$, $\frac{1}{1594323}a^{27}$, $\frac{1}{4782969}a^{28}$, $\frac{1}{4782969}a^{29}$, $\frac{1}{14348907}a^{30}$, $\frac{1}{14348907}a^{31}$, $\frac{1}{43046721}a^{32}$, $\frac{1}{43046721}a^{33}$, $\frac{1}{129140163}a^{34}$, $\frac{1}{129140163}a^{35}$, $\frac{1}{387420489}a^{36}$, $\frac{1}{387420489}a^{37}$, $\frac{1}{1162261467}a^{38}$, $\frac{1}{1162261467}a^{39}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{129140163} a^{34} + \frac{1}{531441} a^{24} + \frac{1}{2187} a^{14} + \frac{1}{9} a^{4} \) (order $50$) | 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
$C_2\times C_{20}$ (as 40T2):
An abelian group of order 40 |
The 40 conjugacy class representatives for $C_2\times C_{20}$ |
Character table for $C_2\times C_{20}$ |
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 | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{10}$ | ${\href{/padicField/11.5.0.1}{5} }^{8}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 $40$ | $2$ | $20$ | $40$ | |||
\(3\) | Deg $40$ | $2$ | $20$ | $20$ | |||
\(5\) | Deg $40$ | $20$ | $2$ | $70$ |