Normalized defining polynomial
\( x^{40} + 5 x^{38} + 20 x^{36} + 75 x^{34} + 275 x^{32} + 1000 x^{30} + 3625 x^{28} + 13125 x^{26} + \cdots + 9765625 \)
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: | \(31654584865659568778929513407372752241664000000000000000000000000000000\) \(\medspace = 2^{40}\cdot 5^{30}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(57.88\) | 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: | $2\cdot 5^{3/4}11^{9/10}\approx 57.87765351369302$ | ||
Ramified primes: | \(2\), \(5\), \(11\) | 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: | \(220=2^{2}\cdot 5\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(3,·)$, $\chi_{220}(7,·)$, $\chi_{220}(9,·)$, $\chi_{220}(141,·)$, $\chi_{220}(147,·)$, $\chi_{220}(149,·)$, $\chi_{220}(23,·)$, $\chi_{220}(127,·)$, $\chi_{220}(27,·)$, $\chi_{220}(29,·)$, $\chi_{220}(161,·)$, $\chi_{220}(163,·)$, $\chi_{220}(167,·)$, $\chi_{220}(129,·)$, $\chi_{220}(41,·)$, $\chi_{220}(43,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(181,·)$, $\chi_{220}(183,·)$, $\chi_{220}(61,·)$, $\chi_{220}(63,·)$, $\chi_{220}(67,·)$, $\chi_{220}(69,·)$, $\chi_{220}(201,·)$, $\chi_{220}(203,·)$, $\chi_{220}(207,·)$, $\chi_{220}(81,·)$, $\chi_{220}(83,·)$, $\chi_{220}(87,·)$, $\chi_{220}(89,·)$, $\chi_{220}(101,·)$, $\chi_{220}(103,·)$, $\chi_{220}(107,·)$, $\chi_{220}(109,·)$, $\chi_{220}(189,·)$, $\chi_{220}(169,·)$, $\chi_{220}(123,·)$, $\chi_{220}(21,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{625}a^{16}$, $\frac{1}{625}a^{17}$, $\frac{1}{625}a^{18}$, $\frac{1}{625}a^{19}$, $\frac{1}{3125}a^{20}$, $\frac{1}{3125}a^{21}$, $\frac{1}{621875}a^{22}+\frac{76}{199}$, $\frac{1}{621875}a^{23}+\frac{76}{199}a$, $\frac{1}{3109375}a^{24}+\frac{55}{199}a^{2}$, $\frac{1}{3109375}a^{25}+\frac{55}{199}a^{3}$, $\frac{1}{3109375}a^{26}+\frac{76}{995}a^{4}$, $\frac{1}{3109375}a^{27}+\frac{76}{995}a^{5}$, $\frac{1}{15546875}a^{28}+\frac{11}{199}a^{6}$, $\frac{1}{15546875}a^{29}+\frac{11}{199}a^{7}$, $\frac{1}{15546875}a^{30}+\frac{76}{4975}a^{8}$, $\frac{1}{15546875}a^{31}+\frac{76}{4975}a^{9}$, $\frac{1}{77734375}a^{32}+\frac{11}{995}a^{10}$, $\frac{1}{77734375}a^{33}+\frac{11}{995}a^{11}$, $\frac{1}{77734375}a^{34}+\frac{76}{24875}a^{12}$, $\frac{1}{77734375}a^{35}+\frac{76}{24875}a^{13}$, $\frac{1}{388671875}a^{36}+\frac{11}{4975}a^{14}$, $\frac{1}{388671875}a^{37}+\frac{11}{4975}a^{15}$, $\frac{1}{388671875}a^{38}+\frac{76}{124375}a^{16}$, $\frac{1}{388671875}a^{39}+\frac{76}{124375}a^{17}$
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: | \( -\frac{76}{388671875} a^{38} - \frac{304}{388671875} a^{36} - \frac{228}{77734375} a^{34} - \frac{836}{77734375} a^{32} - \frac{608}{15546875} a^{30} - \frac{2204}{15546875} a^{28} - \frac{1596}{3109375} a^{26} - \frac{5776}{3109375} a^{24} - \frac{836}{124375} a^{22} - \frac{76}{3125} a^{20} - \frac{11}{125} a^{18} - \frac{5776}{124375} a^{16} - \frac{1596}{24875} a^{14} - \frac{2204}{24875} a^{12} - \frac{608}{4975} a^{10} - \frac{836}{4975} a^{8} - \frac{228}{995} a^{6} - \frac{304}{995} a^{4} - \frac{76}{199} a^{2} - \frac{76}{199} \) (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 | $20^{2}$ | R | $20^{2}$ | R | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{8}$ |
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 $20$ | $2$ | $10$ | $20$ | |||
Deg $20$ | $2$ | $10$ | $20$ | ||||
\(5\) | Deg $20$ | $4$ | $5$ | $15$ | |||
Deg $20$ | $4$ | $5$ | $15$ | ||||
\(11\) | 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |
11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |