Normalized defining polynomial
\( x^{36} + 90 x^{34} + 3645 x^{32} + 88050 x^{30} + 1417950 x^{28} + 16119000 x^{26} + 133603875 x^{24} + \cdots + 17578125 \)
Invariants
| Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4468717346860908762056395509492084398101221888000000000000000000000000000\)
\(\medspace = 2^{36}\cdot 3^{90}\cdot 5^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(104.25\) | 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: | $2\cdot 3^{5/2}5^{3/4}\approx 104.24629667594421$ | ||
| 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: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(540=2^{2}\cdot 3^{3}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{540}(1,·)$, $\chi_{540}(263,·)$, $\chi_{540}(407,·)$, $\chi_{540}(143,·)$, $\chi_{540}(529,·)$, $\chi_{540}(23,·)$, $\chi_{540}(409,·)$, $\chi_{540}(287,·)$, $\chi_{540}(289,·)$, $\chi_{540}(421,·)$, $\chi_{540}(167,·)$, $\chi_{540}(169,·)$, $\chi_{540}(301,·)$, $\chi_{540}(47,·)$, $\chi_{540}(49,·)$, $\chi_{540}(181,·)$, $\chi_{540}(443,·)$, $\chi_{540}(61,·)$, $\chi_{540}(323,·)$, $\chi_{540}(203,·)$, $\chi_{540}(83,·)$, $\chi_{540}(469,·)$, $\chi_{540}(527,·)$, $\chi_{540}(347,·)$, $\chi_{540}(349,·)$, $\chi_{540}(481,·)$, $\chi_{540}(227,·)$, $\chi_{540}(229,·)$, $\chi_{540}(361,·)$, $\chi_{540}(107,·)$, $\chi_{540}(109,·)$, $\chi_{540}(241,·)$, $\chi_{540}(467,·)$, $\chi_{540}(503,·)$, $\chi_{540}(121,·)$, $\chi_{540}(383,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{625}a^{16}$, $\frac{1}{625}a^{17}$, $\frac{1}{1875}a^{18}$, $\frac{1}{1875}a^{19}$, $\frac{1}{9375}a^{20}$, $\frac{1}{9375}a^{21}$, $\frac{1}{9375}a^{22}$, $\frac{1}{9375}a^{23}$, $\frac{1}{46875}a^{24}$, $\frac{1}{46875}a^{25}$, $\frac{1}{46875}a^{26}$, $\frac{1}{46875}a^{27}$, $\frac{1}{234375}a^{28}$, $\frac{1}{234375}a^{29}$, $\frac{1}{234375}a^{30}$, $\frac{1}{234375}a^{31}$, $\frac{1}{505078125}a^{32}-\frac{199}{101015625}a^{30}+\frac{172}{101015625}a^{28}+\frac{52}{6734375}a^{26}-\frac{7}{20203125}a^{24}-\frac{14}{1346875}a^{22}+\frac{58}{1346875}a^{20}-\frac{64}{269375}a^{18}+\frac{18}{269375}a^{16}+\frac{16}{10775}a^{14}+\frac{17}{53875}a^{12}+\frac{119}{10775}a^{10}-\frac{164}{10775}a^{8}+\frac{102}{2155}a^{6}+\frac{2}{431}a^{4}+\frac{83}{431}a^{2}+\frac{145}{431}$, $\frac{1}{505078125}a^{33}-\frac{199}{101015625}a^{31}+\frac{172}{101015625}a^{29}+\frac{52}{6734375}a^{27}-\frac{7}{20203125}a^{25}-\frac{14}{1346875}a^{23}+\frac{58}{1346875}a^{21}-\frac{64}{269375}a^{19}+\frac{18}{269375}a^{17}+\frac{16}{10775}a^{15}+\frac{17}{53875}a^{13}+\frac{119}{10775}a^{11}-\frac{164}{10775}a^{9}+\frac{102}{2155}a^{7}+\frac{2}{431}a^{5}+\frac{83}{431}a^{3}+\frac{145}{431}a$, $\frac{1}{56\!\cdots\!75}a^{34}-\frac{752728}{11\!\cdots\!75}a^{32}+\frac{637092272}{11\!\cdots\!75}a^{30}-\frac{3656342}{374184064765625}a^{28}+\frac{457973209}{74836812953125}a^{26}-\frac{47824727}{224510438859375}a^{24}-\frac{197007572}{44902087771875}a^{22}+\frac{748437637}{14967362590625}a^{20}+\frac{671251816}{8980417554375}a^{18}+\frac{935640359}{2993472518125}a^{16}+\frac{1012807872}{598694503625}a^{14}-\frac{49281216}{23947780145}a^{12}+\frac{2356512101}{119738900725}a^{10}+\frac{2329935526}{119738900725}a^{8}-\frac{2152770664}{23947780145}a^{6}+\frac{169919429}{23947780145}a^{4}+\frac{78972754}{4789556029}a^{2}-\frac{1788688379}{4789556029}$, $\frac{1}{56\!\cdots\!75}a^{35}-\frac{752728}{11\!\cdots\!75}a^{33}+\frac{637092272}{11\!\cdots\!75}a^{31}-\frac{3656342}{374184064765625}a^{29}+\frac{457973209}{74836812953125}a^{27}-\frac{47824727}{224510438859375}a^{25}-\frac{197007572}{44902087771875}a^{23}+\frac{748437637}{14967362590625}a^{21}+\frac{671251816}{8980417554375}a^{19}+\frac{935640359}{2993472518125}a^{17}+\frac{1012807872}{598694503625}a^{15}-\frac{49281216}{23947780145}a^{13}+\frac{2356512101}{119738900725}a^{11}+\frac{2329935526}{119738900725}a^{9}-\frac{2152770664}{23947780145}a^{7}+\frac{169919429}{23947780145}a^{5}+\frac{78972754}{4789556029}a^{3}-\frac{1788688379}{4789556029}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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
| A cyclic group of order 36 |
| The 36 conjugacy class representatives for $C_{36}$ |
| Character table for $C_{36}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.18000.1, 6.6.820125.1, \(\Q(\zeta_{27})^+\), 12.0.3099363912000000000.1, 18.18.1923380668327365689220703125.1 |
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 | $36$ | ${\href{/padicField/11.9.0.1}{9} }^{4}$ | $36$ | ${\href{/padicField/17.12.0.1}{12} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{12}$ | $36$ | ${\href{/padicField/29.9.0.1}{9} }^{4}$ | $18^{2}$ | ${\href{/padicField/37.12.0.1}{12} }^{3}$ | $18^{2}$ | $36$ | $36$ | ${\href{/padicField/53.4.0.1}{4} }^{9}$ | $18^{2}$ |
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 $36$ | $2$ | $18$ | $36$ | |||
|
\(3\)
| Deg $36$ | $18$ | $2$ | $90$ | |||
|
\(5\)
| Deg $36$ | $4$ | $9$ | $27$ |