Normalized defining polynomial
\( x^{9} - x^{8} - 238 x^{7} - 285 x^{6} + 15680 x^{5} + 55764 x^{4} - 208901 x^{3} - 1310052 x^{2} - 1827386 x - 343139 \)
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: | \(165247690404112349401\) \(\medspace = 19^{6}\cdot 37^{8}\) | 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: | \(176.38\) | 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: | $19^{2/3}37^{8/9}\approx 176.3841224409683$ | ||
Ramified primes: | \(19\), \(37\) | 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: | \(703=19\cdot 37\) | ||
Dirichlet character group: | $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(292,·)$, $\chi_{703}(7,·)$, $\chi_{703}(201,·)$, $\chi_{703}(330,·)$, $\chi_{703}(49,·)$, $\chi_{703}(343,·)$, $\chi_{703}(248,·)$, $\chi_{703}(638,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{23}a^{6}+\frac{11}{23}a^{5}-\frac{9}{23}a^{4}+\frac{3}{23}a^{3}+\frac{10}{23}a^{2}-\frac{1}{23}a+\frac{11}{23}$, $\frac{1}{23}a^{7}+\frac{8}{23}a^{5}+\frac{10}{23}a^{4}+\frac{4}{23}a^{2}-\frac{1}{23}a-\frac{6}{23}$, $\frac{1}{41\!\cdots\!11}a^{8}+\frac{48254617162527}{41\!\cdots\!11}a^{7}-\frac{838626189770321}{41\!\cdots\!11}a^{6}+\frac{56\!\cdots\!99}{41\!\cdots\!11}a^{5}-\frac{93\!\cdots\!64}{41\!\cdots\!11}a^{4}-\frac{11\!\cdots\!47}{41\!\cdots\!11}a^{3}+\frac{10\!\cdots\!58}{41\!\cdots\!11}a^{2}-\frac{78\!\cdots\!22}{41\!\cdots\!11}a-\frac{232564917655428}{41\!\cdots\!11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (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: | $\frac{5429443608}{136429331284573}a^{8}-\frac{19390499196}{136429331284573}a^{7}-\frac{1253808385689}{136429331284573}a^{6}+\frac{1745433927252}{136429331284573}a^{5}+\frac{82902824765339}{136429331284573}a^{4}+\frac{80935463803145}{136429331284573}a^{3}-\frac{14\!\cdots\!37}{136429331284573}a^{2}-\frac{143552732372253}{5931710055851}a-\frac{497993322433033}{136429331284573}$, $\frac{1737174800}{136429331284573}a^{8}-\frac{4366854593}{136429331284573}a^{7}-\frac{405865064899}{136429331284573}a^{6}+\frac{139986984207}{136429331284573}a^{5}+\frac{26744444979705}{136429331284573}a^{4}+\frac{52990292145576}{136429331284573}a^{3}-\frac{427850976639639}{136429331284573}a^{2}-\frac{15\!\cdots\!82}{136429331284573}a-\frac{10\!\cdots\!24}{136429331284573}$, $\frac{801490494845}{41\!\cdots\!11}a^{8}+\frac{1628206113851}{41\!\cdots\!11}a^{7}-\frac{254127878057272}{41\!\cdots\!11}a^{6}+\frac{52284668358347}{41\!\cdots\!11}a^{5}+\frac{16\!\cdots\!21}{41\!\cdots\!11}a^{4}+\frac{21\!\cdots\!16}{41\!\cdots\!11}a^{3}-\frac{27\!\cdots\!07}{41\!\cdots\!11}a^{2}-\frac{70\!\cdots\!64}{41\!\cdots\!11}a-\frac{14\!\cdots\!95}{41\!\cdots\!11}$, $\frac{95229275886245}{41\!\cdots\!11}a^{8}-\frac{427791677791928}{41\!\cdots\!11}a^{7}-\frac{21\!\cdots\!90}{41\!\cdots\!11}a^{6}+\frac{46\!\cdots\!03}{41\!\cdots\!11}a^{5}+\frac{13\!\cdots\!90}{41\!\cdots\!11}a^{4}+\frac{66\!\cdots\!92}{41\!\cdots\!11}a^{3}-\frac{22\!\cdots\!36}{41\!\cdots\!11}a^{2}-\frac{47\!\cdots\!38}{41\!\cdots\!11}a-\frac{93\!\cdots\!69}{41\!\cdots\!11}$, $\frac{11033487345372}{41\!\cdots\!11}a^{8}-\frac{67666956296394}{41\!\cdots\!11}a^{7}-\frac{22\!\cdots\!60}{41\!\cdots\!11}a^{6}+\frac{81\!\cdots\!28}{41\!\cdots\!11}a^{5}+\frac{12\!\cdots\!95}{41\!\cdots\!11}a^{4}-\frac{33\!\cdots\!54}{41\!\cdots\!11}a^{3}-\frac{18\!\cdots\!46}{41\!\cdots\!11}a^{2}-\frac{39\!\cdots\!66}{41\!\cdots\!11}a-\frac{16\!\cdots\!33}{41\!\cdots\!11}$, $\frac{5479649104696}{41\!\cdots\!11}a^{8}+\frac{25236428537757}{41\!\cdots\!11}a^{7}-\frac{11\!\cdots\!23}{41\!\cdots\!11}a^{6}-\frac{81\!\cdots\!67}{41\!\cdots\!11}a^{5}+\frac{43\!\cdots\!10}{41\!\cdots\!11}a^{4}+\frac{55\!\cdots\!53}{41\!\cdots\!11}a^{3}+\frac{18\!\cdots\!72}{41\!\cdots\!11}a^{2}+\frac{21\!\cdots\!62}{41\!\cdots\!11}a+\frac{39\!\cdots\!38}{41\!\cdots\!11}$, $\frac{2503450171354}{41\!\cdots\!11}a^{8}-\frac{17456224595367}{41\!\cdots\!11}a^{7}-\frac{537903926239154}{41\!\cdots\!11}a^{6}+\frac{24\!\cdots\!74}{41\!\cdots\!11}a^{5}+\frac{34\!\cdots\!24}{41\!\cdots\!11}a^{4}-\frac{37\!\cdots\!86}{41\!\cdots\!11}a^{3}-\frac{78\!\cdots\!38}{41\!\cdots\!11}a^{2}-\frac{14\!\cdots\!47}{41\!\cdots\!11}a-\frac{28\!\cdots\!73}{41\!\cdots\!11}$, $\frac{4268301291196}{41\!\cdots\!11}a^{8}-\frac{41835080448008}{41\!\cdots\!11}a^{7}-\frac{983313901346462}{41\!\cdots\!11}a^{6}+\frac{58\!\cdots\!45}{41\!\cdots\!11}a^{5}+\frac{71\!\cdots\!50}{41\!\cdots\!11}a^{4}-\frac{44\!\cdots\!24}{41\!\cdots\!11}a^{3}-\frac{13\!\cdots\!89}{41\!\cdots\!11}a^{2}-\frac{24\!\cdots\!37}{41\!\cdots\!11}a-\frac{45\!\cdots\!10}{41\!\cdots\!11}$ (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: | \( 5458915.67791 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 5458915.67791 \cdot 3}{2\cdot\sqrt{165247690404112349401}}\cr\approx \mathstrut & 0.326136875995 \end{aligned}\] (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
3.3.1369.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.1.0.1}{1} }^{9}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.1.0.1}{1} }^{9}$ | R | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.1.0.1}{1} }^{9}$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.9.6.1 | $x^{9} + 1444 x^{3} - 116603$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
\(37\) | 37.9.8.1 | $x^{9} + 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |