Normalized defining polynomial
\( x^{32} - x^{31} + x^{29} - x^{28} + x^{26} - x^{25} + x^{23} - x^{22} + x^{20} - x^{19} + x^{17} + \cdots + 1 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(352701833122210710593389803720131611763844129\) \(\medspace = 3^{16}\cdot 17^{30}\) | 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: | \(24.67\) | 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))
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Galois root discriminant: | $3^{1/2}17^{15/16}\approx 24.666440964016477$ | ||
Ramified primes: | \(3\), \(17\) | 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: | \(51=3\cdot 17\) | ||
Dirichlet character group: | $\lbrace$$\chi_{51}(1,·)$, $\chi_{51}(2,·)$, $\chi_{51}(4,·)$, $\chi_{51}(5,·)$, $\chi_{51}(7,·)$, $\chi_{51}(8,·)$, $\chi_{51}(10,·)$, $\chi_{51}(11,·)$, $\chi_{51}(13,·)$, $\chi_{51}(14,·)$, $\chi_{51}(16,·)$, $\chi_{51}(19,·)$, $\chi_{51}(20,·)$, $\chi_{51}(22,·)$, $\chi_{51}(23,·)$, $\chi_{51}(25,·)$, $\chi_{51}(26,·)$, $\chi_{51}(28,·)$, $\chi_{51}(29,·)$, $\chi_{51}(31,·)$, $\chi_{51}(32,·)$, $\chi_{51}(35,·)$, $\chi_{51}(37,·)$, $\chi_{51}(38,·)$, $\chi_{51}(40,·)$, $\chi_{51}(41,·)$, $\chi_{51}(43,·)$, $\chi_{51}(44,·)$, $\chi_{51}(46,·)$, $\chi_{51}(47,·)$, $\chi_{51}(49,·)$, $\chi_{51}(50,·)$$\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
$C_{5}$, which has order $5$ (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 $102$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $a^{3}+1$, $a^{6}+1$, $a^{21}-a^{19}+a^{6}-a^{2}$, $a^{19}-a^{18}+a^{2}-1$, $a^{6}+a^{3}+1$, $a^{18}+a^{15}+a^{12}+a^{9}+a^{6}+a^{3}+1$, $a^{27}+a^{3}$, $a-1$, $a^{2}-1$, $a^{4}-1$, $a^{8}-1$, $a^{5}-1$, $a^{7}-1$, $a^{11}-1$, $a^{14}-1$ (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)]
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Regulator: | \( 9977645145.96466 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 9977645145.96466 \cdot 5}{102\cdot\sqrt{352701833122210710593389803720131611763844129}}\cr\approx \mathstrut & 0.153663925012584 \end{aligned}\] (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 $C_2\times C_{16}$ |
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 | ${\href{/padicField/2.8.0.1}{8} }^{4}$ | R | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | 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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $32$ | $2$ | $16$ | $16$ | |||
\(17\) | Deg $32$ | $16$ | $2$ | $30$ |