Normalized defining polynomial
\( x^{9} - x^{8} - 84x^{7} + 121x^{6} + 1940x^{5} - 3682x^{4} - 13415x^{3} + 34419x^{2} - 2009x - 26411 \)
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: | \(81976414938366169\) \(\medspace = 13^{6}\cdot 19^{8}\) | 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: | \(75.74\) | 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: | $13^{2/3}19^{8/9}\approx 75.73537306555289$ | ||
Ramified primes: | \(13\), \(19\) | 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: | \(247=13\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(139,·)$, $\chi_{247}(9,·)$, $\chi_{247}(144,·)$, $\chi_{247}(235,·)$, $\chi_{247}(16,·)$, $\chi_{247}(81,·)$, $\chi_{247}(55,·)$, $\chi_{247}(61,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{2}+\frac{3}{7}a$, $\frac{1}{77}a^{6}+\frac{5}{77}a^{5}-\frac{17}{77}a^{4}+\frac{27}{77}a^{3}+\frac{16}{77}a^{2}-\frac{18}{77}a$, $\frac{1}{77}a^{7}+\frac{2}{77}a^{5}-\frac{20}{77}a^{4}+\frac{5}{11}a^{3}+\frac{12}{77}a^{2}-\frac{9}{77}a$, $\frac{1}{68936483}a^{8}+\frac{285256}{68936483}a^{7}+\frac{36811}{9848069}a^{6}-\frac{1818906}{68936483}a^{5}-\frac{22685543}{68936483}a^{4}+\frac{393025}{9848069}a^{3}+\frac{26153320}{68936483}a^{2}+\frac{957108}{9848069}a+\frac{3616}{18271}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{3}$, which has order $3$
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)
| |
Fundamental units: | $\frac{35069}{6266953}a^{8}+\frac{39285}{6266953}a^{7}-\frac{404542}{895279}a^{6}-\frac{1841324}{6266953}a^{5}+\frac{62036204}{6266953}a^{4}+\frac{1243204}{895279}a^{3}-\frac{426549722}{6266953}a^{2}+\frac{30610743}{895279}a+\frac{106763}{1661}$, $\frac{2181}{6266953}a^{8}+\frac{5820}{6266953}a^{7}-\frac{22898}{895279}a^{6}-\frac{378293}{6266953}a^{5}+\frac{2784933}{6266953}a^{4}+\frac{870602}{895279}a^{3}-\frac{10775021}{6266953}a^{2}-\frac{2954155}{895279}a-\frac{1593}{1661}$, $\frac{428649}{68936483}a^{8}+\frac{179161}{68936483}a^{7}-\frac{5154122}{9848069}a^{6}-\frac{615264}{68936483}a^{5}+\frac{852512231}{68936483}a^{4}-\frac{41942301}{9848069}a^{3}-\frac{6493104007}{68936483}a^{2}+\frac{693963906}{9848069}a+\frac{2075664}{18271}$, $\frac{5077}{9848069}a^{8}-\frac{60916}{9848069}a^{7}-\frac{3269}{200981}a^{6}+\frac{4437724}{9848069}a^{5}-\frac{9648161}{9848069}a^{4}-\frac{8575355}{1406867}a^{3}+\frac{195213208}{9848069}a^{2}-\frac{6092461}{1406867}a-\frac{319797}{18271}$, $\frac{68666}{68936483}a^{8}-\frac{420745}{68936483}a^{7}-\frac{603770}{9848069}a^{6}+\frac{31127543}{68936483}a^{5}+\frac{17886249}{68936483}a^{4}-\frac{61609777}{9848069}a^{3}+\frac{703833198}{68936483}a^{2}+\frac{32239243}{9848069}a-\frac{207615}{18271}$, $\frac{12388}{9848069}a^{8}+\frac{85115}{9848069}a^{7}-\frac{120847}{1406867}a^{6}-\frac{6108918}{9848069}a^{5}+\frac{12616910}{9848069}a^{4}+\frac{15668951}{1406867}a^{3}-\frac{28850095}{9848069}a^{2}-\frac{10156200}{200981}a+\frac{71238}{18271}$, $\frac{40122}{68936483}a^{8}-\frac{205504}{68936483}a^{7}-\frac{663099}{9848069}a^{6}+\frac{18426733}{68936483}a^{5}+\frac{168827951}{68936483}a^{4}-\frac{66362211}{9848069}a^{3}-\frac{2032168867}{68936483}a^{2}+\frac{469906504}{9848069}a+\frac{922962}{18271}$, $\frac{5771074}{68936483}a^{8}+\frac{7888929}{68936483}a^{7}-\frac{66592675}{9848069}a^{6}-\frac{404263795}{68936483}a^{5}+\frac{10241881693}{68936483}a^{4}+\frac{419815798}{9848069}a^{3}-\frac{70438255618}{68936483}a^{2}+\frac{4655344846}{9848069}a+\frac{17082590}{18271}$ | 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: | \( 421404.034925 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 421404.034925 \cdot 3}{2\cdot\sqrt{81976414938366169}}\cr\approx \mathstrut & 1.13035653364 \end{aligned}\]
Galois group
A cyclic group of order 9 |
The 9 conjugacy class representatives for $C_9$ |
Character table for $C_9$ |
Intermediate fields
3.3.361.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.1.0.1}{1} }^{9}$ | ${\href{/padicField/11.1.0.1}{1} }^{9}$ | R | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.9.6.3 | $x^{9} + 338 x^{3} - 24167$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
\(19\) | 19.9.8.3 | $x^{9} + 152$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.247.9t1.b.a | $1$ | $ 13 \cdot 19 $ | 9.9.81976414938366169.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.247.9t1.b.b | $1$ | $ 13 \cdot 19 $ | 9.9.81976414938366169.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.247.9t1.b.c | $1$ | $ 13 \cdot 19 $ | 9.9.81976414938366169.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.247.9t1.b.d | $1$ | $ 13 \cdot 19 $ | 9.9.81976414938366169.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.247.9t1.b.e | $1$ | $ 13 \cdot 19 $ | 9.9.81976414938366169.1 | $C_9$ (as 9T1) | $0$ | $1$ |
* | 1.247.9t1.b.f | $1$ | $ 13 \cdot 19 $ | 9.9.81976414938366169.1 | $C_9$ (as 9T1) | $0$ | $1$ |