Normalized defining polynomial
\( x^{40} + 20 x^{38} + 230 x^{36} + 1800 x^{34} + 10625 x^{32} + 49005 x^{30} + 181750 x^{28} + 546975 x^{26} + \cdots + 25 \)
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$$\chi_{300}(1,·)$, $\chi_{300}(7,·)$, $\chi_{300}(269,·)$, $\chi_{300}(143,·)$, $\chi_{300}(149,·)$, $\chi_{300}(23,·)$, $\chi_{300}(281,·)$, $\chi_{300}(283,·)$, $\chi_{300}(29,·)$, $\chi_{300}(287,·)$, $\chi_{300}(161,·)$, $\chi_{300}(163,·)$, $\chi_{300}(167,·)$, $\chi_{300}(41,·)$, $\chi_{300}(43,·)$, $\chi_{300}(47,·)$, $\chi_{300}(49,·)$, $\chi_{300}(181,·)$, $\chi_{300}(187,·)$, $\chi_{300}(61,·)$, $\chi_{300}(67,·)$, $\chi_{300}(289,·)$, $\chi_{300}(203,·)$, $\chi_{300}(209,·)$, $\chi_{300}(83,·)$, $\chi_{300}(89,·)$, $\chi_{300}(221,·)$, $\chi_{300}(223,·)$, $\chi_{300}(263,·)$, $\chi_{300}(227,·)$, $\chi_{300}(101,·)$, $\chi_{300}(103,·)$, $\chi_{300}(107,·)$, $\chi_{300}(109,·)$, $\chi_{300}(229,·)$, $\chi_{300}(241,·)$, $\chi_{300}(169,·)$, $\chi_{300}(121,·)$, $\chi_{300}(127,·)$, $\chi_{300}(247,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5}a^{20}$, $\frac{1}{5}a^{21}$, $\frac{1}{5}a^{22}$, $\frac{1}{5}a^{23}$, $\frac{1}{5}a^{24}$, $\frac{1}{5}a^{25}$, $\frac{1}{5}a^{26}$, $\frac{1}{5}a^{27}$, $\frac{1}{5}a^{28}$, $\frac{1}{5}a^{29}$, $\frac{1}{5}a^{30}$, $\frac{1}{5}a^{31}$, $\frac{1}{5}a^{32}$, $\frac{1}{5}a^{33}$, $\frac{1}{5}a^{34}$, $\frac{1}{5}a^{35}$, $\frac{1}{159215}a^{36}+\frac{8053}{159215}a^{34}+\frac{2127}{159215}a^{32}+\frac{7682}{159215}a^{30}+\frac{2753}{159215}a^{28}+\frac{4261}{159215}a^{26}+\frac{2812}{159215}a^{24}-\frac{1039}{159215}a^{22}+\frac{2399}{31843}a^{20}-\frac{5561}{31843}a^{18}-\frac{1878}{31843}a^{16}-\frac{474}{31843}a^{14}+\frac{3072}{31843}a^{12}+\frac{3375}{31843}a^{10}-\frac{8693}{31843}a^{8}+\frac{1671}{31843}a^{6}+\frac{10754}{31843}a^{4}+\frac{695}{31843}a^{2}-\frac{15756}{31843}$, $\frac{1}{159215}a^{37}+\frac{8053}{159215}a^{35}+\frac{2127}{159215}a^{33}+\frac{7682}{159215}a^{31}+\frac{2753}{159215}a^{29}+\frac{4261}{159215}a^{27}+\frac{2812}{159215}a^{25}-\frac{1039}{159215}a^{23}+\frac{2399}{31843}a^{21}-\frac{5561}{31843}a^{19}-\frac{1878}{31843}a^{17}-\frac{474}{31843}a^{15}+\frac{3072}{31843}a^{13}+\frac{3375}{31843}a^{11}-\frac{8693}{31843}a^{9}+\frac{1671}{31843}a^{7}+\frac{10754}{31843}a^{5}+\frac{695}{31843}a^{3}-\frac{15756}{31843}a$, $\frac{1}{10\!\cdots\!65}a^{38}+\frac{14\!\cdots\!96}{10\!\cdots\!65}a^{36}+\frac{48\!\cdots\!07}{10\!\cdots\!65}a^{34}-\frac{30\!\cdots\!97}{10\!\cdots\!65}a^{32}+\frac{86\!\cdots\!13}{10\!\cdots\!65}a^{30}-\frac{88\!\cdots\!61}{10\!\cdots\!65}a^{28}+\frac{10\!\cdots\!44}{10\!\cdots\!65}a^{26}-\frac{15\!\cdots\!44}{15\!\cdots\!95}a^{24}-\frac{13\!\cdots\!92}{10\!\cdots\!65}a^{22}+\frac{66\!\cdots\!42}{15\!\cdots\!95}a^{20}+\frac{48\!\cdots\!47}{21\!\cdots\!93}a^{18}+\frac{79\!\cdots\!62}{21\!\cdots\!93}a^{16}+\frac{28\!\cdots\!57}{21\!\cdots\!93}a^{14}-\frac{49\!\cdots\!74}{21\!\cdots\!93}a^{12}+\frac{87\!\cdots\!70}{21\!\cdots\!93}a^{10}+\frac{11\!\cdots\!30}{31\!\cdots\!99}a^{8}-\frac{54\!\cdots\!06}{21\!\cdots\!93}a^{6}+\frac{42\!\cdots\!71}{21\!\cdots\!93}a^{4}-\frac{85\!\cdots\!28}{21\!\cdots\!93}a^{2}-\frac{63\!\cdots\!52}{21\!\cdots\!93}$, $\frac{1}{10\!\cdots\!65}a^{39}+\frac{14\!\cdots\!96}{10\!\cdots\!65}a^{37}+\frac{48\!\cdots\!07}{10\!\cdots\!65}a^{35}-\frac{30\!\cdots\!97}{10\!\cdots\!65}a^{33}+\frac{86\!\cdots\!13}{10\!\cdots\!65}a^{31}-\frac{88\!\cdots\!61}{10\!\cdots\!65}a^{29}+\frac{10\!\cdots\!44}{10\!\cdots\!65}a^{27}-\frac{15\!\cdots\!44}{15\!\cdots\!95}a^{25}-\frac{13\!\cdots\!92}{10\!\cdots\!65}a^{23}+\frac{66\!\cdots\!42}{15\!\cdots\!95}a^{21}+\frac{48\!\cdots\!47}{21\!\cdots\!93}a^{19}+\frac{79\!\cdots\!62}{21\!\cdots\!93}a^{17}+\frac{28\!\cdots\!57}{21\!\cdots\!93}a^{15}-\frac{49\!\cdots\!74}{21\!\cdots\!93}a^{13}+\frac{87\!\cdots\!70}{21\!\cdots\!93}a^{11}+\frac{11\!\cdots\!30}{31\!\cdots\!99}a^{9}-\frac{54\!\cdots\!06}{21\!\cdots\!93}a^{7}+\frac{42\!\cdots\!71}{21\!\cdots\!93}a^{5}-\frac{85\!\cdots\!28}{21\!\cdots\!93}a^{3}-\frac{63\!\cdots\!52}{21\!\cdots\!93}a$
| 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: |
\( \frac{34771395730757978124566515565983}{10876794422132693343948145156820465} a^{38} + \frac{692283897168111053725510776099378}{10876794422132693343948145156820465} a^{36} + \frac{1587052547053860559766113339320657}{2175358884426538668789629031364093} a^{34} + \frac{12375918588351769666644002884613743}{2175358884426538668789629031364093} a^{32} + \frac{363948310199062342368789764166523708}{10876794422132693343948145156820465} a^{30} + \frac{334362934312097178180994489480061962}{2175358884426538668789629031364093} a^{28} + \frac{1234580133632388750749612083207465799}{2175358884426538668789629031364093} a^{26} + \frac{3696145712529795454860576250672591245}{2175358884426538668789629031364093} a^{24} + \frac{9045892737790323647042167510486708285}{2175358884426538668789629031364093} a^{22} + \frac{90123004371087405788337777342915478407}{10876794422132693343948145156820465} a^{20} + \frac{29158312147958218840435273548763128420}{2175358884426538668789629031364093} a^{18} + \frac{37698071349679749583427711324594752322}{2175358884426538668789629031364093} a^{16} + \frac{38528234138397276629821371218314527235}{2175358884426538668789629031364093} a^{14} + \frac{30166374776853215627973237685016843565}{2175358884426538668789629031364093} a^{12} + \frac{17933471660986256714922124117907747132}{2175358884426538668789629031364093} a^{10} + \frac{7607606223328870595462112891904849700}{2175358884426538668789629031364093} a^{8} + \frac{2324457873859495166602237549634081505}{2175358884426538668789629031364093} a^{6} + \frac{427937480517537119948273759350450550}{2175358884426538668789629031364093} a^{4} + \frac{59899867790215038918631270886861075}{2175358884426538668789629031364093} a^{2} + \frac{471995768929960333928148596728894}{310765554918076952684232718766299} \)
(order $6$)
| 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 $C_2\times C_{20}$ is not computed |
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 | ${\href{/padicField/7.4.0.1}{4} }^{10}$ | ${\href{/padicField/11.10.0.1}{10} }^{4}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{8}$ | $20^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 $20$ | $2$ | $10$ | $20$ | |||
| Deg $20$ | $2$ | $10$ | $20$ | ||||
|
\(3\)
| Deg $40$ | $2$ | $20$ | $20$ | |||
|
\(5\)
| Deg $40$ | $20$ | $2$ | $70$ |