Normalized defining polynomial
\( x^{40} + 4 x^{38} + 14 x^{36} + 48 x^{34} + 164 x^{32} + 560 x^{30} + 1912 x^{28} + 6528 x^{26} + \cdots + 1024 \)
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: | \(40126953504928778716867347648517224229883090465646395672541598175985664\) \(\medspace = 2^{110}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(58.22\) | 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: | $2^{11/4}11^{9/10}\approx 58.22183708777889$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(176=2^{4}\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(5,·)$, $\chi_{176}(129,·)$, $\chi_{176}(9,·)$, $\chi_{176}(13,·)$, $\chi_{176}(45,·)$, $\chi_{176}(17,·)$, $\chi_{176}(21,·)$, $\chi_{176}(153,·)$, $\chi_{176}(25,·)$, $\chi_{176}(157,·)$, $\chi_{176}(133,·)$, $\chi_{176}(161,·)$, $\chi_{176}(37,·)$, $\chi_{176}(41,·)$, $\chi_{176}(173,·)$, $\chi_{176}(29,·)$, $\chi_{176}(49,·)$, $\chi_{176}(53,·)$, $\chi_{176}(137,·)$, $\chi_{176}(57,·)$, $\chi_{176}(61,·)$, $\chi_{176}(65,·)$, $\chi_{176}(69,·)$, $\chi_{176}(73,·)$, $\chi_{176}(141,·)$, $\chi_{176}(81,·)$, $\chi_{176}(85,·)$, $\chi_{176}(89,·)$, $\chi_{176}(93,·)$, $\chi_{176}(97,·)$, $\chi_{176}(101,·)$, $\chi_{176}(145,·)$, $\chi_{176}(105,·)$, $\chi_{176}(109,·)$, $\chi_{176}(113,·)$, $\chi_{176}(117,·)$, $\chi_{176}(169,·)$, $\chi_{176}(125,·)$, $\chi_{176}(149,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{32}a^{20}$, $\frac{1}{32}a^{21}$, $\frac{1}{259808}a^{22}-\frac{3363}{8119}$, $\frac{1}{259808}a^{23}-\frac{3363}{8119}a$, $\frac{1}{519616}a^{24}+\frac{2378}{8119}a^{2}$, $\frac{1}{519616}a^{25}+\frac{2378}{8119}a^{3}$, $\frac{1}{519616}a^{26}-\frac{3363}{16238}a^{4}$, $\frac{1}{519616}a^{27}-\frac{3363}{16238}a^{5}$, $\frac{1}{1039232}a^{28}+\frac{1189}{8119}a^{6}$, $\frac{1}{1039232}a^{29}+\frac{1189}{8119}a^{7}$, $\frac{1}{1039232}a^{30}-\frac{3363}{32476}a^{8}$, $\frac{1}{1039232}a^{31}-\frac{3363}{32476}a^{9}$, $\frac{1}{2078464}a^{32}+\frac{1189}{16238}a^{10}$, $\frac{1}{2078464}a^{33}+\frac{1189}{16238}a^{11}$, $\frac{1}{2078464}a^{34}-\frac{3363}{64952}a^{12}$, $\frac{1}{2078464}a^{35}-\frac{3363}{64952}a^{13}$, $\frac{1}{4156928}a^{36}+\frac{1189}{32476}a^{14}$, $\frac{1}{4156928}a^{37}+\frac{1189}{32476}a^{15}$, $\frac{1}{4156928}a^{38}-\frac{3363}{129904}a^{16}$, $\frac{1}{4156928}a^{39}-\frac{3363}{129904}a^{17}$
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)
| |
Torsion generator: | \( \frac{51}{519616} a^{38} - \frac{9369319}{129904} a^{16} \) (order $22$) | 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 | $20^{2}$ | $20^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | R | $20^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | $20^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{8}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | ${\href{/padicField/47.5.0.1}{5} }^{8}$ | $20^{2}$ | $20^{2}$ |
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 $40$ | $4$ | $10$ | $110$ | |||
\(11\) | Deg $40$ | $10$ | $4$ | $36$ |