Normalized defining polynomial
\( x^{32} - x^{31} + 6 x^{30} - 46 x^{29} + 391 x^{28} + 717 x^{27} + 7428 x^{26} + 20280 x^{25} + \cdots + 272803795483 \)
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: |
\(9572290431813101812896740520288704359886188011817648342612467121075247600488097\)
\(\medspace = 353^{31}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(293.87\) | 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: | $353^{31/32}\approx 293.8710101725786$ | ||
| Ramified primes: |
\(353\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{353}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(353\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{353}(1,·)$, $\chi_{353}(6,·)$, $\chi_{353}(7,·)$, $\chi_{353}(137,·)$, $\chi_{353}(10,·)$, $\chi_{353}(283,·)$, $\chi_{353}(286,·)$, $\chi_{353}(36,·)$, $\chi_{353}(293,·)$, $\chi_{353}(294,·)$, $\chi_{353}(42,·)$, $\chi_{353}(304,·)$, $\chi_{353}(49,·)$, $\chi_{353}(311,·)$, $\chi_{353}(59,·)$, $\chi_{353}(60,·)$, $\chi_{353}(317,·)$, $\chi_{353}(67,·)$, $\chi_{353}(70,·)$, $\chi_{353}(343,·)$, $\chi_{353}(216,·)$, $\chi_{353}(346,·)$, $\chi_{353}(347,·)$, $\chi_{353}(352,·)$, $\chi_{353}(100,·)$, $\chi_{353}(101,·)$, $\chi_{353}(106,·)$, $\chi_{353}(237,·)$, $\chi_{353}(116,·)$, $\chi_{353}(247,·)$, $\chi_{353}(252,·)$, $\chi_{353}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{262}a^{29}-\frac{33}{262}a^{28}-\frac{14}{131}a^{27}+\frac{15}{131}a^{25}+\frac{9}{262}a^{24}+\frac{58}{131}a^{23}-\frac{31}{131}a^{22}+\frac{17}{131}a^{21}-\frac{56}{131}a^{20}+\frac{79}{262}a^{19}-\frac{47}{262}a^{18}-\frac{47}{131}a^{17}+\frac{51}{131}a^{16}+\frac{2}{131}a^{15}+\frac{62}{131}a^{14}+\frac{7}{131}a^{13}+\frac{85}{262}a^{12}+\frac{10}{131}a^{11}+\frac{83}{262}a^{10}+\frac{127}{262}a^{9}-\frac{107}{262}a^{8}-\frac{9}{131}a^{7}+\frac{69}{262}a^{6}+\frac{9}{131}a^{5}-\frac{43}{131}a^{4}-\frac{123}{262}a^{3}-\frac{31}{262}a^{2}+\frac{19}{262}a+\frac{60}{131}$, $\frac{1}{262}a^{30}+\frac{31}{131}a^{28}-\frac{7}{262}a^{27}+\frac{15}{131}a^{26}-\frac{49}{262}a^{25}+\frac{10}{131}a^{24}+\frac{49}{131}a^{23}+\frac{42}{131}a^{22}+\frac{93}{262}a^{21}+\frac{51}{262}a^{20}-\frac{30}{131}a^{19}-\frac{73}{262}a^{18}-\frac{59}{131}a^{17}-\frac{18}{131}a^{16}+\frac{125}{262}a^{15}+\frac{45}{262}a^{14}+\frac{23}{262}a^{13}+\frac{37}{131}a^{12}+\frac{44}{131}a^{11}+\frac{115}{262}a^{10}-\frac{54}{131}a^{9}+\frac{119}{262}a^{8}+\frac{65}{131}a^{7}-\frac{63}{262}a^{6}-\frac{8}{131}a^{5}-\frac{79}{262}a^{4}-\frac{29}{262}a^{3}+\frac{22}{131}a^{2}+\frac{46}{131}a+\frac{15}{131}$, $\frac{1}{53\!\cdots\!78}a^{31}-\frac{18\!\cdots\!51}{15\!\cdots\!94}a^{30}+\frac{83\!\cdots\!47}{53\!\cdots\!78}a^{29}+\frac{27\!\cdots\!04}{26\!\cdots\!89}a^{28}-\frac{10\!\cdots\!22}{26\!\cdots\!89}a^{27}-\frac{28\!\cdots\!75}{53\!\cdots\!78}a^{26}-\frac{56\!\cdots\!31}{26\!\cdots\!89}a^{25}-\frac{23\!\cdots\!35}{26\!\cdots\!89}a^{24}+\frac{31\!\cdots\!29}{53\!\cdots\!78}a^{23}+\frac{10\!\cdots\!41}{53\!\cdots\!78}a^{22}-\frac{21\!\cdots\!99}{53\!\cdots\!78}a^{21}-\frac{13\!\cdots\!29}{29\!\cdots\!94}a^{20}+\frac{10\!\cdots\!47}{53\!\cdots\!78}a^{19}-\frac{25\!\cdots\!85}{53\!\cdots\!78}a^{18}-\frac{82\!\cdots\!08}{26\!\cdots\!89}a^{17}-\frac{12\!\cdots\!22}{26\!\cdots\!89}a^{16}+\frac{11\!\cdots\!71}{53\!\cdots\!78}a^{15}+\frac{21\!\cdots\!05}{53\!\cdots\!78}a^{14}-\frac{77\!\cdots\!05}{26\!\cdots\!89}a^{13}+\frac{77\!\cdots\!81}{53\!\cdots\!78}a^{12}-\frac{12\!\cdots\!89}{53\!\cdots\!78}a^{11}-\frac{26\!\cdots\!13}{53\!\cdots\!78}a^{10}-\frac{85\!\cdots\!81}{26\!\cdots\!89}a^{9}-\frac{26\!\cdots\!81}{53\!\cdots\!78}a^{8}-\frac{57\!\cdots\!38}{26\!\cdots\!89}a^{7}-\frac{16\!\cdots\!77}{53\!\cdots\!78}a^{6}-\frac{35\!\cdots\!05}{53\!\cdots\!78}a^{5}+\frac{66\!\cdots\!89}{53\!\cdots\!78}a^{4}+\frac{16\!\cdots\!65}{53\!\cdots\!78}a^{3}-\frac{81\!\cdots\!58}{26\!\cdots\!89}a^{2}-\frac{20\!\cdots\!49}{53\!\cdots\!78}a+\frac{61\!\cdots\!95}{26\!\cdots\!89}$
| 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
| \(\Q(\sqrt{353}) \), 4.4.43986977.1, 8.8.683003513396280737.1, 16.16.164672311157017194379126294334593898657.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{4}$ | $32$ | $32$ | $32$ | ${\href{/padicField/11.8.0.1}{8} }^{4}$ | $32$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{4}$ | $32$ | $32$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $32$ | $32$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(353\)
| Deg $32$ | $32$ | $1$ | $31$ |