Normalized defining polynomial
\( x^{9} - x^{8} - 56x^{7} + 118x^{6} + 573x^{5} - 1249x^{4} - 1582x^{3} + 2700x^{2} + 1576x - 32 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(67675234241018881\) \(\medspace = 127^{8}\) | 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: | \(74.14\) | 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: | $127^{8/9}\approx 74.13917550324533$ | ||
Ramified primes: | \(127\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $9$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(127\) | ||
Dirichlet character group: | $\lbrace$$\chi_{127}(1,·)$, $\chi_{127}(99,·)$, $\chi_{127}(68,·)$, $\chi_{127}(37,·)$, $\chi_{127}(103,·)$, $\chi_{127}(107,·)$, $\chi_{127}(19,·)$, $\chi_{127}(52,·)$, $\chi_{127}(22,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{128}a^{7}-\frac{1}{32}a^{6}+\frac{1}{64}a^{5}-\frac{1}{16}a^{4}-\frac{15}{128}a^{3}+\frac{7}{32}a^{2}+\frac{3}{32}a-\frac{1}{8}$, $\frac{1}{33536}a^{8}-\frac{5}{16768}a^{7}+\frac{17}{16768}a^{6}-\frac{47}{8384}a^{5}-\frac{1927}{33536}a^{4}+\frac{1759}{16768}a^{3}-\frac{975}{8384}a^{2}-\frac{2087}{4192}a-\frac{233}{1048}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{1}{1048}a^{8}-\frac{211}{8384}a^{7}-\frac{325}{2096}a^{6}+\frac{4357}{4192}a^{5}+\frac{861}{262}a^{4}-\frac{34867}{8384}a^{3}-\frac{29153}{2096}a^{2}-\frac{13873}{2096}a+\frac{71}{524}$, $\frac{61}{524}a^{8}-\frac{721}{8384}a^{7}-\frac{13581}{2096}a^{6}+\frac{50915}{4192}a^{5}+\frac{34937}{524}a^{4}-\frac{1081609}{8384}a^{3}-\frac{388431}{2096}a^{2}+\frac{569733}{2096}a+\frac{90013}{524}$, $\frac{101}{16768}a^{8}-\frac{243}{8384}a^{7}-\frac{2475}{8384}a^{6}+\frac{7567}{4192}a^{5}+\frac{4493}{16768}a^{4}-\frac{101895}{8384}a^{3}+\frac{19425}{4192}a^{2}+\frac{43615}{2096}a-\frac{215}{524}$, $\frac{129}{4192}a^{8}+\frac{151}{4192}a^{7}-\frac{3309}{2096}a^{6}+\frac{843}{2096}a^{5}+\frac{66865}{4192}a^{4}-\frac{20529}{4192}a^{3}-\frac{19985}{524}a^{2}-\frac{2263}{1048}a+\frac{15}{262}$, $\frac{5431}{16768}a^{8}+\frac{1553}{4192}a^{7}-\frac{145045}{8384}a^{6}+\frac{315}{262}a^{5}+\frac{3116543}{16768}a^{4}-\frac{29629}{4192}a^{3}-\frac{2104851}{4192}a^{2}-\frac{201077}{1048}a+\frac{3710}{131}$, $\frac{18}{131}a^{8}-\frac{123}{8384}a^{7}-\frac{17063}{2096}a^{6}+\frac{29393}{4192}a^{5}+\frac{26651}{262}a^{4}-\frac{223235}{8384}a^{3}-\frac{649373}{2096}a^{2}-\frac{285673}{2096}a+\frac{1447}{524}$, $\frac{2515}{16768}a^{8}+\frac{2883}{8384}a^{7}-\frac{60997}{8384}a^{6}-\frac{26243}{4192}a^{5}+\frac{1102043}{16768}a^{4}+\frac{241055}{8384}a^{3}-\frac{607121}{4192}a^{2}-\frac{151071}{2096}a+\frac{623}{524}$, $\frac{4801}{4192}a^{8}+\frac{2735}{1048}a^{7}-\frac{116455}{2096}a^{6}-\frac{24595}{524}a^{5}+\frac{2102505}{4192}a^{4}+\frac{221543}{1048}a^{3}-\frac{1164935}{1048}a^{2}-\frac{71302}{131}a+\frac{1417}{131}$ (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: | \( 64259015.2682 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 64259015.2682 \cdot 1}{2\cdot\sqrt{67675234241018881}}\cr\approx \mathstrut & 63.2352365416 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 9 |
The 9 conjugacy class representatives for $C_9$ |
Character table for $C_9$ |
Intermediate fields
3.3.16129.1 |
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 | ${\href{/padicField/2.1.0.1}{1} }^{9}$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(127\) | 127.9.8.1 | $x^{9} + 127$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.127.9t1.a.a | $1$ | $ 127 $ | 9.9.67675234241018881.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.127.9t1.a.b | $1$ | $ 127 $ | 9.9.67675234241018881.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.127.3t1.a.a | $1$ | $ 127 $ | 3.3.16129.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.127.9t1.a.c | $1$ | $ 127 $ | 9.9.67675234241018881.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.127.9t1.a.d | $1$ | $ 127 $ | 9.9.67675234241018881.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.127.3t1.a.b | $1$ | $ 127 $ | 3.3.16129.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.127.9t1.a.e | $1$ | $ 127 $ | 9.9.67675234241018881.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.127.9t1.a.f | $1$ | $ 127 $ | 9.9.67675234241018881.1 | $C_9$ (as 9T1) | $0$ | $1$ |