Normalized defining polynomial
\( x^{40} - x^{39} + 5 x^{38} - 5 x^{37} + 20 x^{36} - 19 x^{35} + 74 x^{34} - 65 x^{33} + 265 x^{32} - 210 x^{31} + 936 x^{30} - 1475 x^{29} + 4114 x^{28} - 6060 x^{27} + 15680 x^{26} - 21989 x^{25} + \cdots + 1 \)
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: | \(100383397447978918530459891214693626269465146363712847232818603515625\) \(\medspace = 3^{20}\cdot 5^{30}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(50.12\) | 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: | $3^{1/2}5^{3/4}11^{9/10}\approx 50.12351825429183$ | ||
Ramified primes: | \(3\), \(5\), \(11\) | 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: | \(165=3\cdot 5\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(4,·)$, $\chi_{165}(8,·)$, $\chi_{165}(137,·)$, $\chi_{165}(139,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(19,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(34,·)$, $\chi_{165}(38,·)$, $\chi_{165}(46,·)$, $\chi_{165}(47,·)$, $\chi_{165}(49,·)$, $\chi_{165}(53,·)$, $\chi_{165}(151,·)$, $\chi_{165}(61,·)$, $\chi_{165}(62,·)$, $\chi_{165}(64,·)$, $\chi_{165}(68,·)$, $\chi_{165}(76,·)$, $\chi_{165}(79,·)$, $\chi_{165}(83,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(122,·)$, $\chi_{165}(94,·)$, $\chi_{165}(98,·)$, $\chi_{165}(106,·)$, $\chi_{165}(107,·)$, $\chi_{165}(109,·)$, $\chi_{165}(113,·)$, $\chi_{165}(136,·)$, $\chi_{165}(124,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{25\!\cdots\!21}a^{31}+\frac{969862058041036}{25\!\cdots\!21}a^{30}+\frac{241303777751570}{25\!\cdots\!21}a^{29}+\frac{11\!\cdots\!57}{25\!\cdots\!21}a^{28}+\frac{12\!\cdots\!45}{25\!\cdots\!21}a^{27}+\frac{710382005693106}{25\!\cdots\!21}a^{26}+\frac{743525622606569}{25\!\cdots\!21}a^{25}+\frac{759727505551736}{25\!\cdots\!21}a^{24}-\frac{12\!\cdots\!23}{25\!\cdots\!21}a^{23}-\frac{10\!\cdots\!26}{25\!\cdots\!21}a^{22}+\frac{10\!\cdots\!30}{25\!\cdots\!21}a^{21}+\frac{95503815941798}{25\!\cdots\!21}a^{20}+\frac{25370031947195}{25\!\cdots\!21}a^{19}-\frac{528579085352603}{25\!\cdots\!21}a^{18}-\frac{70364038799755}{25\!\cdots\!21}a^{17}+\frac{488230004469246}{25\!\cdots\!21}a^{16}-\frac{383166537792952}{25\!\cdots\!21}a^{15}+\frac{57003490590598}{25\!\cdots\!21}a^{14}+\frac{433614414914333}{25\!\cdots\!21}a^{13}+\frac{299709387039926}{25\!\cdots\!21}a^{12}-\frac{480277222460171}{25\!\cdots\!21}a^{11}+\frac{830362944453688}{25\!\cdots\!21}a^{10}-\frac{14059251761729}{25\!\cdots\!21}a^{9}-\frac{484226564561075}{25\!\cdots\!21}a^{8}+\frac{126960133769841}{25\!\cdots\!21}a^{7}-\frac{424260348281824}{25\!\cdots\!21}a^{6}-\frac{990746123121383}{25\!\cdots\!21}a^{5}+\frac{285771829634526}{25\!\cdots\!21}a^{4}-\frac{10\!\cdots\!53}{25\!\cdots\!21}a^{3}+\frac{11\!\cdots\!70}{25\!\cdots\!21}a^{2}-\frac{572259025431674}{25\!\cdots\!21}a+\frac{808763874066416}{25\!\cdots\!21}$, $\frac{1}{25\!\cdots\!21}a^{32}+\frac{12\!\cdots\!25}{25\!\cdots\!21}a^{30}-\frac{613072852992611}{25\!\cdots\!21}a^{29}+\frac{503379744866868}{25\!\cdots\!21}a^{28}-\frac{539158269730133}{25\!\cdots\!21}a^{27}-\frac{11\!\cdots\!64}{25\!\cdots\!21}a^{26}-\frac{920953358335506}{25\!\cdots\!21}a^{25}-\frac{12\!\cdots\!08}{25\!\cdots\!21}a^{24}+\frac{11\!\cdots\!24}{25\!\cdots\!21}a^{23}-\frac{10\!\cdots\!53}{25\!\cdots\!21}a^{22}-\frac{95503815942622}{25\!\cdots\!21}a^{21}+\frac{209142208610269}{25\!\cdots\!21}a^{20}+\frac{639739986408395}{25\!\cdots\!21}a^{19}-\frac{693236095240533}{25\!\cdots\!21}a^{18}-\frac{868143299208031}{25\!\cdots\!21}a^{17}+\frac{445016655269210}{25\!\cdots\!21}a^{16}+\frac{248343766226902}{25\!\cdots\!21}a^{15}-\frac{12\!\cdots\!53}{25\!\cdots\!21}a^{14}-\frac{163212758531710}{25\!\cdots\!21}a^{13}+\frac{773526171669338}{25\!\cdots\!21}a^{12}+\frac{913440026709071}{25\!\cdots\!21}a^{11}-\frac{304794668595476}{25\!\cdots\!21}a^{10}-\frac{916762311003590}{25\!\cdots\!21}a^{9}-\frac{10\!\cdots\!44}{25\!\cdots\!21}a^{8}+\frac{23573903759522}{25\!\cdots\!21}a^{7}-\frac{201973183291819}{25\!\cdots\!21}a^{6}-\frac{655525585245758}{25\!\cdots\!21}a^{5}+\frac{10\!\cdots\!14}{25\!\cdots\!21}a^{4}+\frac{558223872986398}{25\!\cdots\!21}a^{3}+\frac{11\!\cdots\!45}{25\!\cdots\!21}a^{2}+\frac{814362117403119}{25\!\cdots\!21}a+\frac{61996354910397}{25\!\cdots\!21}$, $\frac{1}{25\!\cdots\!21}a^{33}+\frac{936795771304776}{25\!\cdots\!21}a^{22}-\frac{554016545945649}{25\!\cdots\!21}a^{11}+\frac{204895633579702}{25\!\cdots\!21}$, $\frac{1}{25\!\cdots\!21}a^{34}+\frac{936795771304776}{25\!\cdots\!21}a^{23}-\frac{554016545945649}{25\!\cdots\!21}a^{12}+\frac{204895633579702}{25\!\cdots\!21}a$, $\frac{1}{25\!\cdots\!21}a^{35}+\frac{936795771304776}{25\!\cdots\!21}a^{24}-\frac{554016545945649}{25\!\cdots\!21}a^{13}+\frac{204895633579702}{25\!\cdots\!21}a^{2}$, $\frac{1}{25\!\cdots\!21}a^{36}+\frac{936795771304776}{25\!\cdots\!21}a^{25}-\frac{554016545945649}{25\!\cdots\!21}a^{14}+\frac{204895633579702}{25\!\cdots\!21}a^{3}$, $\frac{1}{25\!\cdots\!21}a^{37}+\frac{936795771304776}{25\!\cdots\!21}a^{26}-\frac{554016545945649}{25\!\cdots\!21}a^{15}+\frac{204895633579702}{25\!\cdots\!21}a^{4}$, $\frac{1}{25\!\cdots\!21}a^{38}+\frac{936795771304776}{25\!\cdots\!21}a^{27}-\frac{554016545945649}{25\!\cdots\!21}a^{16}+\frac{204895633579702}{25\!\cdots\!21}a^{5}$, $\frac{1}{25\!\cdots\!21}a^{39}+\frac{936795771304776}{25\!\cdots\!21}a^{28}-\frac{554016545945649}{25\!\cdots\!21}a^{17}+\frac{204895633579702}{25\!\cdots\!21}a^{6}$
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: | \( \frac{1008753747971}{2526205995232921} a^{39} - \frac{831212771199384}{2526205995232921} a^{28} - \frac{1248337057599278259}{2526205995232921} a^{17} - \frac{30260868834152543254}{2526205995232921} a^{6} \) (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 $C_2\times C_{20}$ 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 | $20^{2}$ | R | R | $20^{2}$ | R | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{8}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.20.10.2 | $x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$ | $2$ | $10$ | $10$ | 20T1 | $[\ ]_{2}^{10}$ |
3.20.10.2 | $x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$ | $2$ | $10$ | $10$ | 20T1 | $[\ ]_{2}^{10}$ | |
\(5\) | 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 | $[\ ]_{10}^{2}$ |