Normalized defining polynomial
\( x^{36} + 72 x^{34} + 2268 x^{32} + 41280 x^{30} + 483804 x^{28} + 3859488 x^{26} + 21644784 x^{24} + \cdots + 512 \)
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: |
\(614667125325361522818798575155151578949632894783197825857500612833312768\)
\(\medspace = 2^{99}\cdot 3^{88}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(98.66\) | 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: | $2^{11/4}3^{22/9}\approx 98.65722338828127$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(432=2^{4}\cdot 3^{3}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{432}(1,·)$, $\chi_{432}(259,·)$, $\chi_{432}(385,·)$, $\chi_{432}(265,·)$, $\chi_{432}(139,·)$, $\chi_{432}(115,·)$, $\chi_{432}(145,·)$, $\chi_{432}(19,·)$, $\chi_{432}(25,·)$, $\chi_{432}(409,·)$, $\chi_{432}(283,·)$, $\chi_{432}(289,·)$, $\chi_{432}(163,·)$, $\chi_{432}(169,·)$, $\chi_{432}(427,·)$, $\chi_{432}(49,·)$, $\chi_{432}(307,·)$, $\chi_{432}(313,·)$, $\chi_{432}(187,·)$, $\chi_{432}(193,·)$, $\chi_{432}(67,·)$, $\chi_{432}(73,·)$, $\chi_{432}(331,·)$, $\chi_{432}(337,·)$, $\chi_{432}(211,·)$, $\chi_{432}(43,·)$, $\chi_{432}(217,·)$, $\chi_{432}(91,·)$, $\chi_{432}(97,·)$, $\chi_{432}(355,·)$, $\chi_{432}(361,·)$, $\chi_{432}(235,·)$, $\chi_{432}(241,·)$, $\chi_{432}(403,·)$, $\chi_{432}(121,·)$, $\chi_{432}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{32}a^{20}$, $\frac{1}{32}a^{21}$, $\frac{1}{32}a^{22}$, $\frac{1}{32}a^{23}$, $\frac{1}{64}a^{24}$, $\frac{1}{64}a^{25}$, $\frac{1}{64}a^{26}$, $\frac{1}{64}a^{27}$, $\frac{1}{128}a^{28}$, $\frac{1}{128}a^{29}$, $\frac{1}{128}a^{30}$, $\frac{1}{128}a^{31}$, $\frac{1}{256}a^{32}$, $\frac{1}{256}a^{33}$, $\frac{1}{29\!\cdots\!04}a^{34}+\frac{75\!\cdots\!95}{74\!\cdots\!76}a^{32}-\frac{26\!\cdots\!43}{14\!\cdots\!52}a^{30}+\frac{38\!\cdots\!33}{14\!\cdots\!52}a^{28}-\frac{13\!\cdots\!25}{18\!\cdots\!44}a^{26}-\frac{20\!\cdots\!25}{74\!\cdots\!76}a^{24}-\frac{27\!\cdots\!13}{18\!\cdots\!44}a^{22}+\frac{74\!\cdots\!51}{93\!\cdots\!72}a^{20}+\frac{69\!\cdots\!37}{18\!\cdots\!44}a^{18}-\frac{47\!\cdots\!27}{23\!\cdots\!18}a^{16}+\frac{28\!\cdots\!65}{46\!\cdots\!36}a^{14}-\frac{31\!\cdots\!26}{11\!\cdots\!09}a^{12}+\frac{28\!\cdots\!97}{46\!\cdots\!36}a^{10}+\frac{19\!\cdots\!79}{23\!\cdots\!18}a^{8}+\frac{51\!\cdots\!81}{23\!\cdots\!18}a^{6}-\frac{26\!\cdots\!85}{23\!\cdots\!18}a^{4}+\frac{33\!\cdots\!04}{11\!\cdots\!09}a^{2}-\frac{50\!\cdots\!45}{11\!\cdots\!09}$, $\frac{1}{29\!\cdots\!04}a^{35}+\frac{75\!\cdots\!95}{74\!\cdots\!76}a^{33}-\frac{26\!\cdots\!43}{14\!\cdots\!52}a^{31}+\frac{38\!\cdots\!33}{14\!\cdots\!52}a^{29}-\frac{13\!\cdots\!25}{18\!\cdots\!44}a^{27}-\frac{20\!\cdots\!25}{74\!\cdots\!76}a^{25}-\frac{27\!\cdots\!13}{18\!\cdots\!44}a^{23}+\frac{74\!\cdots\!51}{93\!\cdots\!72}a^{21}+\frac{69\!\cdots\!37}{18\!\cdots\!44}a^{19}-\frac{47\!\cdots\!27}{23\!\cdots\!18}a^{17}+\frac{28\!\cdots\!65}{46\!\cdots\!36}a^{15}-\frac{31\!\cdots\!26}{11\!\cdots\!09}a^{13}+\frac{28\!\cdots\!97}{46\!\cdots\!36}a^{11}+\frac{19\!\cdots\!79}{23\!\cdots\!18}a^{9}+\frac{51\!\cdots\!81}{23\!\cdots\!18}a^{7}-\frac{26\!\cdots\!85}{23\!\cdots\!18}a^{5}+\frac{33\!\cdots\!04}{11\!\cdots\!09}a^{3}-\frac{50\!\cdots\!45}{11\!\cdots\!09}a$
| 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 $C_{36}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 4.0.2048.2, 6.6.3359232.1, \(\Q(\zeta_{27})^+\), 12.0.369768517790072832.1, 18.18.132173713091594538512566714368.1 |
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 | $36$ | ${\href{/padicField/7.9.0.1}{9} }^{4}$ | $36$ | $36$ | ${\href{/padicField/17.3.0.1}{3} }^{12}$ | ${\href{/padicField/19.12.0.1}{12} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{4}$ | $36$ | $18^{2}$ | ${\href{/padicField/37.12.0.1}{12} }^{3}$ | $18^{2}$ | $36$ | $18^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{9}$ | $36$ |
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 $36$ | $4$ | $9$ | $99$ | |||
|
\(3\)
| Deg $36$ | $9$ | $4$ | $88$ |