Normalized defining polynomial
\( x^{32} - x^{31} + 100 x^{30} - 369 x^{29} + 4310 x^{28} - 25796 x^{27} + 146725 x^{26} + \cdots + 38241539870119 \)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3063491324247901158147277924095178749079926649475132560984974419980910475284097\)
\(\medspace = 3^{16}\cdot 193^{31}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(283.59\) | 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: | $3^{1/2}193^{31/32}\approx 283.5921926498609$ | ||
| Ramified primes: |
\(3\), \(193\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{193}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(579=3\cdot 193\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{579}(512,·)$, $\chi_{579}(1,·)$, $\chi_{579}(260,·)$, $\chi_{579}(385,·)$, $\chi_{579}(8,·)$, $\chi_{579}(14,·)$, $\chi_{579}(529,·)$, $\chi_{579}(274,·)$, $\chi_{579}(23,·)$, $\chi_{579}(202,·)$, $\chi_{579}(410,·)$, $\chi_{579}(166,·)$, $\chi_{579}(170,·)$, $\chi_{579}(43,·)$, $\chi_{579}(428,·)$, $\chi_{579}(179,·)$, $\chi_{579}(436,·)$, $\chi_{579}(184,·)$, $\chi_{579}(185,·)$, $\chi_{579}(314,·)$, $\chi_{579}(317,·)$, $\chi_{579}(190,·)$, $\chi_{579}(64,·)$, $\chi_{579}(322,·)$, $\chi_{579}(196,·)$, $\chi_{579}(455,·)$, $\chi_{579}(458,·)$, $\chi_{579}(343,·)$, $\chi_{579}(344,·)$, $\chi_{579}(220,·)$, $\chi_{579}(362,·)$, $\chi_{579}(112,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{277}a^{29}-\frac{97}{277}a^{28}-\frac{89}{277}a^{27}+\frac{17}{277}a^{26}+\frac{114}{277}a^{25}+\frac{60}{277}a^{24}-\frac{10}{277}a^{23}-\frac{45}{277}a^{22}+\frac{83}{277}a^{21}+\frac{109}{277}a^{20}-\frac{77}{277}a^{19}-\frac{74}{277}a^{18}-\frac{39}{277}a^{17}-\frac{16}{277}a^{16}+\frac{97}{277}a^{15}+\frac{135}{277}a^{14}-\frac{20}{277}a^{13}+\frac{18}{277}a^{12}-\frac{21}{277}a^{11}+\frac{120}{277}a^{10}-\frac{20}{277}a^{9}+\frac{103}{277}a^{8}-\frac{75}{277}a^{7}-\frac{53}{277}a^{6}-\frac{115}{277}a^{5}+\frac{59}{277}a^{4}+\frac{116}{277}a^{3}-\frac{1}{277}a^{2}-\frac{40}{277}a-\frac{114}{277}$, $\frac{1}{26065423}a^{30}-\frac{26533}{26065423}a^{29}+\frac{8571502}{26065423}a^{28}+\frac{11800460}{26065423}a^{27}-\frac{3293257}{26065423}a^{26}+\frac{5323225}{26065423}a^{25}+\frac{6346556}{26065423}a^{24}+\frac{3964204}{26065423}a^{23}-\frac{9061790}{26065423}a^{22}-\frac{2725088}{26065423}a^{21}+\frac{9532708}{26065423}a^{20}+\frac{10996725}{26065423}a^{19}+\frac{404194}{26065423}a^{18}-\frac{4064981}{26065423}a^{17}-\frac{9084675}{26065423}a^{16}-\frac{9206617}{26065423}a^{15}-\frac{11655895}{26065423}a^{14}+\frac{6283690}{26065423}a^{13}+\frac{245716}{26065423}a^{12}-\frac{5215742}{26065423}a^{11}+\frac{11480683}{26065423}a^{10}-\frac{9115763}{26065423}a^{9}-\frac{4919039}{26065423}a^{8}-\frac{8534766}{26065423}a^{7}+\frac{7422696}{26065423}a^{6}+\frac{4874216}{26065423}a^{5}+\frac{4006430}{26065423}a^{4}-\frac{11799556}{26065423}a^{3}+\frac{9743279}{26065423}a^{2}+\frac{7727486}{26065423}a-\frac{11410240}{26065423}$, $\frac{1}{33\!\cdots\!49}a^{31}-\frac{17\!\cdots\!35}{33\!\cdots\!49}a^{30}+\frac{35\!\cdots\!67}{33\!\cdots\!49}a^{29}-\frac{79\!\cdots\!76}{33\!\cdots\!49}a^{28}-\frac{54\!\cdots\!12}{33\!\cdots\!49}a^{27}+\frac{51\!\cdots\!50}{33\!\cdots\!49}a^{26}+\frac{15\!\cdots\!75}{33\!\cdots\!49}a^{25}-\frac{14\!\cdots\!47}{33\!\cdots\!49}a^{24}-\frac{91\!\cdots\!81}{33\!\cdots\!49}a^{23}+\frac{15\!\cdots\!21}{33\!\cdots\!49}a^{22}-\frac{15\!\cdots\!04}{33\!\cdots\!49}a^{21}+\frac{16\!\cdots\!42}{33\!\cdots\!49}a^{20}-\frac{16\!\cdots\!71}{33\!\cdots\!49}a^{19}-\frac{13\!\cdots\!91}{33\!\cdots\!49}a^{18}-\frac{92\!\cdots\!41}{33\!\cdots\!49}a^{17}-\frac{15\!\cdots\!88}{33\!\cdots\!49}a^{16}-\frac{98\!\cdots\!75}{33\!\cdots\!49}a^{15}-\frac{12\!\cdots\!75}{33\!\cdots\!49}a^{14}+\frac{13\!\cdots\!85}{33\!\cdots\!49}a^{13}+\frac{15\!\cdots\!65}{33\!\cdots\!49}a^{12}-\frac{13\!\cdots\!24}{33\!\cdots\!49}a^{11}+\frac{57\!\cdots\!90}{33\!\cdots\!49}a^{10}+\frac{12\!\cdots\!55}{33\!\cdots\!49}a^{9}-\frac{12\!\cdots\!44}{33\!\cdots\!49}a^{8}+\frac{12\!\cdots\!35}{33\!\cdots\!49}a^{7}-\frac{84\!\cdots\!82}{33\!\cdots\!49}a^{6}-\frac{50\!\cdots\!12}{33\!\cdots\!49}a^{5}+\frac{13\!\cdots\!34}{33\!\cdots\!49}a^{4}+\frac{63\!\cdots\!09}{33\!\cdots\!49}a^{3}-\frac{73\!\cdots\!29}{33\!\cdots\!49}a^{2}+\frac{88\!\cdots\!24}{33\!\cdots\!49}a+\frac{73\!\cdots\!04}{33\!\cdots\!49}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $15$ | 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 32 |
| The 32 conjugacy class representatives for $C_{32}$ |
| Character table for $C_{32}$ |
Intermediate fields
| \(\Q(\sqrt{193}) \), 4.4.7189057.1, 8.8.9974730326005057.1, 16.16.19202582299769315484813817587637057.1 |
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 | $16^{2}$ | R | $32$ | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | $32$ | $32$ | $32$ | $32$ | $16^{2}$ | $32$ | $16^{2}$ | $32$ | $32$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | $32$ | $32$ | ${\href{/padicField/59.8.0.1}{8} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
| 3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
|
\(193\)
| Deg $32$ | $32$ | $1$ | $31$ |