Normalized defining polynomial
\( x^{32} - 2 x^{30} + 4 x^{28} - 8 x^{26} + 16 x^{24} - 32 x^{22} + 64 x^{20} - 128 x^{18} + 256 x^{16} + \cdots + 65536 \)
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: | \(2306255574353269294321282113641705264626385702354944\) \(\medspace = 2^{48}\cdot 17^{30}\) | 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: | \(40.28\) | 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: | $2^{3/2}17^{15/16}\approx 40.28012942155011$ | ||
Ramified primes: | \(2\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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: | \(136=2^{3}\cdot 17\) | ||
Dirichlet character group: | $\lbrace$$\chi_{136}(1,·)$, $\chi_{136}(3,·)$, $\chi_{136}(129,·)$, $\chi_{136}(9,·)$, $\chi_{136}(11,·)$, $\chi_{136}(19,·)$, $\chi_{136}(25,·)$, $\chi_{136}(27,·)$, $\chi_{136}(33,·)$, $\chi_{136}(35,·)$, $\chi_{136}(41,·)$, $\chi_{136}(43,·)$, $\chi_{136}(49,·)$, $\chi_{136}(57,·)$, $\chi_{136}(59,·)$, $\chi_{136}(65,·)$, $\chi_{136}(67,·)$, $\chi_{136}(73,·)$, $\chi_{136}(75,·)$, $\chi_{136}(81,·)$, $\chi_{136}(83,·)$, $\chi_{136}(89,·)$, $\chi_{136}(91,·)$, $\chi_{136}(97,·)$, $\chi_{136}(99,·)$, $\chi_{136}(131,·)$, $\chi_{136}(105,·)$, $\chi_{136}(107,·)$, $\chi_{136}(113,·)$, $\chi_{136}(115,·)$, $\chi_{136}(121,·)$, $\chi_{136}(123,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{24}$, which has order $72$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1}{4096} a^{24} \) (order $34$) | 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: | $\frac{1}{4}a^{4}+1$, $\frac{1}{256}a^{16}+\frac{1}{16}a^{8}+1$, $\frac{1}{64}a^{12}+1$, $\frac{1}{2}a^{2}-1$, $\frac{1}{16}a^{8}+1$, $\frac{1}{32768}a^{30}-\frac{1}{16384}a^{28}-\frac{1}{1024}a^{20}-\frac{1}{64}a^{12}-\frac{1}{4}a^{4}$, $\frac{1}{1024}a^{20}-\frac{1}{32}a^{10}$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}$, $\frac{1}{32768}a^{30}-\frac{1}{16384}a^{28}+\frac{1}{8192}a^{26}-\frac{1}{4096}a^{24}+\frac{1}{2048}a^{22}+\frac{1}{1024}a^{21}-\frac{1}{1024}a^{20}+\frac{1}{512}a^{18}-\frac{1}{256}a^{16}+\frac{1}{128}a^{14}-\frac{1}{64}a^{12}+\frac{1}{16}a^{10}-\frac{1}{16}a^{8}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}+\frac{1}{2}a^{2}-1$, $\frac{1}{512}a^{18}+\frac{1}{16}a^{9}+1$, $\frac{1}{64}a^{12}+\frac{1}{8}a^{7}+\frac{1}{2}a^{2}$, $\frac{1}{16384}a^{28}+\frac{1}{128}a^{15}+\frac{1}{2}a^{2}$, $\frac{1}{32768}a^{30}-\frac{1}{32}a^{10}+\frac{1}{2}a^{3}$, $\frac{1}{1024}a^{20}+\frac{1}{256}a^{17}+\frac{1}{128}a^{14}$, $\frac{1}{16384}a^{29}+\frac{1}{128}a^{14}-\frac{1}{32}a^{10}$ (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: | \( 598124168304.8442 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 598124168304.8442 \cdot 72}{34\cdot\sqrt{2306255574353269294321282113641705264626385702354944}}\cr\approx \mathstrut & 0.155621324660132 \end{aligned}\] (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 $16$ | $2$ | $8$ | $24$ | |||
Deg $16$ | $2$ | $8$ | $24$ | ||||
\(17\) | 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |