Normalized defining polynomial
\( x^{32} + 193 x^{30} + 15440 x^{28} + 671061 x^{26} + 17579019 x^{24} + 293049077 x^{22} + \cdots + 780515353 \)
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: | \(305\!\cdots\!872\) \(\medspace = 2^{32}\cdot 193^{31}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(327.46\) | 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: | $2\cdot 193^{31/32}\approx 327.46405753294687$ | ||
Ramified primes: | \(2\), \(193\) | 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: | \(772=2^{2}\cdot 193\) | ||
Dirichlet character group: | $\lbrace$$\chi_{772}(1,·)$, $\chi_{772}(363,·)$, $\chi_{772}(389,·)$, $\chi_{772}(769,·)$, $\chi_{772}(9,·)$, $\chi_{772}(651,·)$, $\chi_{772}(529,·)$, $\chi_{772}(23,·)$, $\chi_{772}(413,·)$, $\chi_{772}(129,·)$, $\chi_{772}(555,·)$, $\chi_{772}(257,·)$, $\chi_{772}(429,·)$, $\chi_{772}(385,·)$, $\chi_{772}(305,·)$, $\chi_{772}(179,·)$, $\chi_{772}(571,·)$, $\chi_{772}(151,·)$, $\chi_{772}(319,·)$, $\chi_{772}(67,·)$, $\chi_{772}(455,·)$, $\chi_{772}(587,·)$, $\chi_{772}(207,·)$, $\chi_{772}(81,·)$, $\chi_{772}(729,·)$, $\chi_{772}(603,·)$, $\chi_{772}(745,·)$, $\chi_{772}(235,·)$, $\chi_{772}(629,·)$, $\chi_{772}(377,·)$, $\chi_{772}(507,·)$, $\chi_{772}(703,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{7}a^{24}+\frac{1}{7}a^{22}+\frac{1}{7}a^{20}-\frac{2}{7}a^{16}+\frac{1}{7}a^{14}-\frac{1}{7}a^{10}-\frac{3}{7}a^{8}+\frac{3}{7}a^{6}-\frac{1}{7}a^{4}+\frac{2}{7}a^{2}+\frac{1}{7}$, $\frac{1}{7}a^{25}+\frac{1}{7}a^{23}+\frac{1}{7}a^{21}-\frac{2}{7}a^{17}+\frac{1}{7}a^{15}-\frac{1}{7}a^{11}-\frac{3}{7}a^{9}+\frac{3}{7}a^{7}-\frac{1}{7}a^{5}+\frac{2}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{7}a^{26}-\frac{1}{7}a^{20}-\frac{2}{7}a^{18}+\frac{3}{7}a^{16}-\frac{1}{7}a^{14}-\frac{1}{7}a^{12}-\frac{2}{7}a^{10}-\frac{1}{7}a^{8}+\frac{3}{7}a^{6}+\frac{3}{7}a^{4}-\frac{1}{7}a^{2}-\frac{1}{7}$, $\frac{1}{7}a^{27}-\frac{1}{7}a^{21}-\frac{2}{7}a^{19}+\frac{3}{7}a^{17}-\frac{1}{7}a^{15}-\frac{1}{7}a^{13}-\frac{2}{7}a^{11}-\frac{1}{7}a^{9}+\frac{3}{7}a^{7}+\frac{3}{7}a^{5}-\frac{1}{7}a^{3}-\frac{1}{7}a$, $\frac{1}{763}a^{28}-\frac{33}{763}a^{26}+\frac{4}{763}a^{24}+\frac{157}{763}a^{22}+\frac{27}{109}a^{20}+\frac{328}{763}a^{18}-\frac{129}{763}a^{16}+\frac{43}{763}a^{14}-\frac{228}{763}a^{12}-\frac{114}{763}a^{10}+\frac{59}{763}a^{8}-\frac{13}{109}a^{6}-\frac{83}{763}a^{4}+\frac{110}{763}a^{2}-\frac{264}{763}$, $\frac{1}{763}a^{29}-\frac{33}{763}a^{27}+\frac{4}{763}a^{25}+\frac{157}{763}a^{23}+\frac{27}{109}a^{21}+\frac{328}{763}a^{19}-\frac{129}{763}a^{17}+\frac{43}{763}a^{15}-\frac{228}{763}a^{13}-\frac{114}{763}a^{11}+\frac{59}{763}a^{9}-\frac{13}{109}a^{7}-\frac{83}{763}a^{5}+\frac{110}{763}a^{3}-\frac{264}{763}a$, $\frac{1}{59\!\cdots\!99}a^{30}-\frac{46\!\cdots\!51}{59\!\cdots\!99}a^{28}-\frac{77\!\cdots\!22}{59\!\cdots\!99}a^{26}+\frac{36\!\cdots\!93}{59\!\cdots\!99}a^{24}+\frac{11\!\cdots\!11}{59\!\cdots\!99}a^{22}+\frac{24\!\cdots\!84}{59\!\cdots\!99}a^{20}-\frac{12\!\cdots\!69}{84\!\cdots\!57}a^{18}-\frac{16\!\cdots\!00}{59\!\cdots\!99}a^{16}-\frac{24\!\cdots\!05}{59\!\cdots\!99}a^{14}-\frac{25\!\cdots\!87}{59\!\cdots\!99}a^{12}+\frac{22\!\cdots\!28}{59\!\cdots\!99}a^{10}+\frac{23\!\cdots\!96}{59\!\cdots\!99}a^{8}-\frac{84\!\cdots\!98}{59\!\cdots\!99}a^{6}-\frac{20\!\cdots\!24}{59\!\cdots\!99}a^{4}+\frac{17\!\cdots\!86}{59\!\cdots\!99}a^{2}-\frac{21\!\cdots\!26}{59\!\cdots\!99}$, $\frac{1}{11\!\cdots\!89}a^{31}+\frac{66\!\cdots\!26}{11\!\cdots\!89}a^{29}-\frac{55\!\cdots\!07}{11\!\cdots\!89}a^{27}+\frac{37\!\cdots\!85}{11\!\cdots\!89}a^{25}+\frac{46\!\cdots\!96}{17\!\cdots\!27}a^{23}-\frac{81\!\cdots\!07}{17\!\cdots\!27}a^{21}+\frac{26\!\cdots\!96}{11\!\cdots\!89}a^{19}-\frac{51\!\cdots\!96}{11\!\cdots\!89}a^{17}-\frac{31\!\cdots\!71}{11\!\cdots\!89}a^{15}+\frac{17\!\cdots\!26}{17\!\cdots\!27}a^{13}-\frac{61\!\cdots\!16}{11\!\cdots\!89}a^{11}-\frac{50\!\cdots\!04}{17\!\cdots\!27}a^{9}+\frac{40\!\cdots\!34}{11\!\cdots\!89}a^{7}-\frac{42\!\cdots\!50}{17\!\cdots\!27}a^{5}-\frac{15\!\cdots\!09}{11\!\cdots\!89}a^{3}-\frac{13\!\cdots\!83}{11\!\cdots\!89}a$
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 $C_{32}$ is not computed |
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 | R | ${\href{/padicField/3.8.0.1}{8} }^{4}$ | $32$ | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | $32$ | $32$ | $32$ | $32$ | $16^{2}$ | $32$ | $16^{2}$ | $32$ | $32$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | $32$ | $32$ | ${\href{/padicField/59.8.0.1}{8} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $2$ | $8$ | $16$ | |||
Deg $16$ | $2$ | $8$ | $16$ | ||||
\(193\) | Deg $32$ | $32$ | $1$ | $31$ |