Normalized defining polynomial
\( x^{32} - 3x^{16} + 3 \)
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: | \(210183365271596281474544229277581939111301912914296832\) \(\medspace = 2^{128}\cdot 3^{31}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(46.38\) | 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: | $2^{597/128}3^{31/32}\approx 73.49096154480738$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
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
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( a^{16} - 1 \) (order $6$) | 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: | $a-1$, $a+1$, $a^{8}+1$, $a^{20}-a^{16}-a^{4}+1$, $a^{28}+a^{24}-a^{16}-2a^{12}-a^{8}+a^{4}+1$, $a^{28}-a^{26}+a^{22}-a^{20}-a^{14}-a^{12}+2a^{10}-a^{8}-a^{6}+2a^{4}-a^{2}+1$, $a^{28}-a^{24}+a^{22}+a^{20}-a^{18}-a^{16}+a^{14}-2a^{12}-a^{10}+2a^{8}-a^{6}-2a^{4}+a^{2}+2$, $a^{31}-a^{30}+a^{28}-a^{27}+a^{25}-a^{24}+a^{22}-a^{21}+a^{19}-a^{18}+a^{16}-2a^{15}+a^{14}+a^{13}-2a^{12}+a^{11}+a^{10}-2a^{9}+a^{8}+a^{7}-2a^{6}+a^{5}+a^{4}-2a^{3}+a^{2}+a-1$, $a^{28}-a^{26}-a^{22}+a^{16}-a^{12}+2a^{10}+2a^{6}-a^{4}-a^{2}-2$, $a^{26}+a^{25}+a^{20}+2a^{16}-a^{11}-2a^{10}-a^{9}-3a^{4}-4$, $2a^{31}+a^{30}+a^{29}-a^{26}-a^{25}-2a^{24}-2a^{23}-2a^{22}+2a^{17}+2a^{16}-4a^{15}-2a^{14}-2a^{13}+2a^{10}+2a^{9}+4a^{8}+4a^{7}+5a^{6}+a^{5}-a^{3}-4a-5$, $2a^{31}+a^{30}-a^{29}-a^{28}+2a^{26}-a^{25}-2a^{24}+a^{23}+a^{18}-2a^{17}-2a^{15}-3a^{14}+a^{13}+2a^{12}+a^{11}-3a^{10}+4a^{8}-a^{6}-2a^{2}+2a+2$, $a^{29}-a^{28}-a^{27}-a^{26}+a^{25}+a^{24}-a^{21}-a^{20}+a^{18}+a^{16}-a^{15}-2a^{13}+2a^{12}+2a^{11}+a^{10}-a^{9}-2a^{8}+a^{5}+a-2$, $a^{31}-3a^{30}+2a^{29}+2a^{28}-3a^{27}+4a^{25}-3a^{24}-2a^{23}+4a^{22}-a^{21}-4a^{20}+4a^{19}+a^{18}-4a^{17}+2a^{16}+a^{15}+4a^{14}-6a^{13}-a^{12}+5a^{11}-3a^{10}-5a^{9}+7a^{8}-6a^{6}+5a^{5}+4a^{4}-8a^{3}+3a^{2}+5a-7$, $4a^{31}-3a^{30}-4a^{27}+5a^{26}-6a^{25}+5a^{24}-4a^{23}+3a^{22}+2a^{21}-2a^{20}+6a^{19}-7a^{18}+8a^{17}-6a^{16}-8a^{15}+8a^{14}-4a^{13}+5a^{12}+3a^{11}-6a^{10}+7a^{9}-11a^{8}+9a^{7}-10a^{6}+3a^{5}-2a^{4}-5a^{3}+10a^{2}-13a+13$ (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: | \( 258430049161748.3 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 258430049161748.3 \cdot 1}{6\cdot\sqrt{210183365271596281474544229277581939111301912914296832}}\cr\approx \mathstrut & 0.554332680322487 \end{aligned}\] (assuming GRH)
Galois group
$D_{32}:C_8$ (as 32T22195):
A solvable group of order 512 |
The 47 conjugacy class representatives for $D_{32}:C_8$ |
Character table for $D_{32}:C_8$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.432.1, 8.0.143327232.1, 16.0.4038858263648368852992.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Arithmetically equvalently sibling: | 32.0.210183365271596281474544229277581939111301912914296832.2 |
Minimal sibling: | 32.0.210183365271596281474544229277581939111301912914296832.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | $32$ | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $16^{2}$ | $16^{2}$ | $32$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{4}$ | $32$ | ${\href{/padicField/31.2.0.1}{2} }^{15}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | $32$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | $32$ | $16^{2}$ |
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 $32$ | $16$ | $2$ | $128$ | |||
\(3\) | Deg $32$ | $32$ | $1$ | $31$ |