Normalized defining polynomial
\( x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
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)
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Discriminant: | \(35190667333271321019306672876612934335729762304\) \(\medspace = 2^{32}\cdot 17^{30}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(28.48\) | 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: | $2\cdot 17^{15/16}\approx 28.48235266104985$ | ||
Ramified primes: | \(2\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(68=2^{2}\cdot 17\) | ||
Dirichlet character group: | $\lbrace$$\chi_{68}(1,·)$, $\chi_{68}(3,·)$, $\chi_{68}(5,·)$, $\chi_{68}(7,·)$, $\chi_{68}(9,·)$, $\chi_{68}(11,·)$, $\chi_{68}(13,·)$, $\chi_{68}(15,·)$, $\chi_{68}(19,·)$, $\chi_{68}(21,·)$, $\chi_{68}(23,·)$, $\chi_{68}(25,·)$, $\chi_{68}(27,·)$, $\chi_{68}(29,·)$, $\chi_{68}(31,·)$, $\chi_{68}(33,·)$, $\chi_{68}(35,·)$, $\chi_{68}(37,·)$, $\chi_{68}(39,·)$, $\chi_{68}(41,·)$, $\chi_{68}(43,·)$, $\chi_{68}(45,·)$, $\chi_{68}(47,·)$, $\chi_{68}(49,·)$, $\chi_{68}(53,·)$, $\chi_{68}(55,·)$, $\chi_{68}(57,·)$, $\chi_{68}(59,·)$, $\chi_{68}(61,·)$, $\chi_{68}(63,·)$, $\chi_{68}(65,·)$, $\chi_{68}(67,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{8}$, which has order $8$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( a \) (order $68$) | 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: | $a^{19}-a^{17}$, $a^{4}+1$, $a^{17}+a$, $a^{8}+a^{4}+1$, $a^{31}+a^{27}+a^{23}-a^{21}+a^{19}-a^{17}+a^{15}-a^{13}-a^{9}-a^{5}-a$, $a^{11}-a^{9}+a^{7}$, $a^{21}+a^{17}+a^{13}+a^{9}+a^{5}$, $a-1$, $a^{3}-1$, $a^{9}-1$, $a^{7}-1$, $a^{11}-1$, $a^{13}-1$, $a^{5}-1$, $a^{15}-1$ (assuming GRH) | 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: | \( 36938367173.77557 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 36938367173.77557 \cdot 8}{68\cdot\sqrt{35190667333271321019306672876612934335729762304}}\cr\approx \mathstrut & 0.136685646313966 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{16}$ (as 32T32):
An abelian group of order 32 |
The 32 conjugacy class representatives for $C_2\times C_{16}$ |
Character table for $C_2\times C_{16}$ is not computed |
Intermediate fields
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 | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{4}$ |
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 $16$ | $2$ | $8$ | $16$ | ||||
\(17\) | 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |