Normalized defining polynomial
\( x^{9} - x^{8} - 959 x^{7} + 270 x^{6} + 130894 x^{5} + 1765845 x^{4} + 11378151 x^{3} + 46627247 x^{2} + \cdots + 147434209 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-165041226467028122193059592704\) \(\medspace = -\,2^{9}\cdot 7^{3}\cdot 2659^{7}\) | 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: | \(1763.60\) | 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: | $2^{3/2}7^{1/2}2659^{5/6}\approx 5345.963155697544$ | ||
Ramified primes: | \(2\), \(7\), \(2659\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-37226}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{14}a^{6}+\frac{2}{7}a^{5}+\frac{3}{14}a^{4}+\frac{2}{7}a^{3}+\frac{2}{7}a-\frac{5}{14}$, $\frac{1}{42}a^{7}+\frac{5}{14}a^{5}+\frac{1}{7}a^{4}-\frac{1}{21}a^{3}+\frac{2}{21}a^{2}-\frac{1}{2}a-\frac{4}{21}$, $\frac{1}{49\!\cdots\!34}a^{8}+\frac{30\!\cdots\!01}{49\!\cdots\!34}a^{7}-\frac{31\!\cdots\!17}{16\!\cdots\!78}a^{6}-\frac{94\!\cdots\!67}{23\!\cdots\!54}a^{5}-\frac{27\!\cdots\!92}{24\!\cdots\!17}a^{4}-\frac{83\!\cdots\!38}{24\!\cdots\!17}a^{3}+\frac{77\!\cdots\!57}{49\!\cdots\!34}a^{2}+\frac{20\!\cdots\!67}{49\!\cdots\!34}a+\frac{79\!\cdots\!93}{24\!\cdots\!17}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{19}\times C_{27949}$, which has order $531031$ (assuming GRH)
Unit group
Rank: | $5$ | 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: | $\frac{147405768631}{35\!\cdots\!31}a^{8}+\frac{1361643036820}{35\!\cdots\!31}a^{7}-\frac{51106852173286}{11\!\cdots\!77}a^{6}-\frac{458586077509106}{11\!\cdots\!77}a^{5}+\frac{29\!\cdots\!99}{35\!\cdots\!31}a^{4}+\frac{43\!\cdots\!30}{35\!\cdots\!31}a^{3}+\frac{29\!\cdots\!92}{35\!\cdots\!31}a^{2}+\frac{95\!\cdots\!30}{35\!\cdots\!31}a+\frac{17\!\cdots\!87}{35\!\cdots\!31}$, $\frac{150440658271}{35\!\cdots\!31}a^{8}+\frac{1393811233948}{35\!\cdots\!31}a^{7}-\frac{52120625255800}{11\!\cdots\!77}a^{6}-\frac{468494616435194}{11\!\cdots\!77}a^{5}+\frac{30\!\cdots\!62}{35\!\cdots\!31}a^{4}+\frac{44\!\cdots\!24}{35\!\cdots\!31}a^{3}+\frac{29\!\cdots\!08}{35\!\cdots\!31}a^{2}+\frac{99\!\cdots\!32}{35\!\cdots\!31}a+\frac{18\!\cdots\!32}{35\!\cdots\!31}$, $\frac{1451165499256}{24\!\cdots\!17}a^{8}+\frac{19925705682952}{24\!\cdots\!17}a^{7}-\frac{149694196824299}{23\!\cdots\!54}a^{6}-\frac{926350717120997}{11\!\cdots\!77}a^{5}+\frac{69\!\cdots\!01}{49\!\cdots\!34}a^{4}+\frac{48\!\cdots\!55}{24\!\cdots\!17}a^{3}+\frac{33\!\cdots\!79}{24\!\cdots\!17}a^{2}+\frac{10\!\cdots\!42}{24\!\cdots\!17}a+\frac{58\!\cdots\!01}{70\!\cdots\!62}$, $\frac{25363480550564}{24\!\cdots\!17}a^{8}-\frac{229314317265371}{49\!\cdots\!34}a^{7}-\frac{16\!\cdots\!59}{16\!\cdots\!78}a^{6}+\frac{90\!\cdots\!81}{23\!\cdots\!54}a^{5}+\frac{66\!\cdots\!69}{49\!\cdots\!34}a^{4}+\frac{31\!\cdots\!52}{24\!\cdots\!17}a^{3}+\frac{13\!\cdots\!62}{24\!\cdots\!17}a^{2}+\frac{66\!\cdots\!97}{49\!\cdots\!34}a+\frac{56\!\cdots\!59}{49\!\cdots\!34}$, $\frac{17\!\cdots\!41}{35\!\cdots\!31}a^{8}+\frac{11\!\cdots\!28}{24\!\cdots\!17}a^{7}-\frac{42\!\cdots\!26}{81\!\cdots\!39}a^{6}-\frac{38\!\cdots\!70}{81\!\cdots\!39}a^{5}+\frac{24\!\cdots\!61}{24\!\cdots\!17}a^{4}+\frac{36\!\cdots\!34}{24\!\cdots\!17}a^{3}+\frac{24\!\cdots\!61}{24\!\cdots\!17}a^{2}+\frac{80\!\cdots\!02}{24\!\cdots\!17}a+\frac{14\!\cdots\!70}{24\!\cdots\!17}$ (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: | \( 328977.09053380624 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 328977.09053380624 \cdot 531031}{2\cdot\sqrt{165041226467028122193059592704}}\cr\approx \mathstrut & 0.426666853654511 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3$ (as 9T4):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
3.3.7070281.1, 3.1.148904.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | data not computed |
Minimal sibling: | 6.0.23342951498961373981184.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.7 | $x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(7\) | 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(2659\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $6$ | $6$ | $1$ | $5$ |