Normalized defining polynomial
\( x^{9} - 18057 x^{7} - 475501 x^{6} + 146478384 x^{5} - 1403751180 x^{4} - 46876525748 x^{3} + \cdots - 55\!\cdots\!28 \)
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: | \(-17971044653397089488481063155542528\) \(\medspace = -\,2^{9}\cdot 3^{13}\cdot 13^{6}\cdot 463^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(6398.29\) | 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: | $2^{3/2}3^{3/2}13^{2/3}463^{5/6}\approx 13526.121045889007$ | ||
Ramified primes: | \(2\), \(3\), \(13\), \(463\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2778}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{13}a^{3}$, $\frac{1}{13}a^{4}$, $\frac{1}{26}a^{5}-\frac{1}{26}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{156494}a^{6}-\frac{1}{26}a^{4}-\frac{1}{26}a^{3}$, $\frac{1}{938964}a^{7}-\frac{1}{52}a^{5}+\frac{5}{156}a^{4}+\frac{1}{3}a$, $\frac{1}{16\!\cdots\!88}a^{8}+\frac{26\!\cdots\!39}{55\!\cdots\!96}a^{7}+\frac{10\!\cdots\!69}{55\!\cdots\!96}a^{6}-\frac{12\!\cdots\!79}{13\!\cdots\!26}a^{5}-\frac{14\!\cdots\!17}{54\!\cdots\!52}a^{4}+\frac{49\!\cdots\!92}{23\!\cdots\!21}a^{3}-\frac{16\!\cdots\!90}{53\!\cdots\!51}a^{2}+\frac{49\!\cdots\!25}{17\!\cdots\!17}a+\frac{85\!\cdots\!60}{31\!\cdots\!81}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{311766}$, which has order $75759138$ (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: | $\frac{15\!\cdots\!35}{41\!\cdots\!97}a^{8}+\frac{48\!\cdots\!33}{14\!\cdots\!64}a^{7}-\frac{10\!\cdots\!23}{27\!\cdots\!98}a^{6}-\frac{14\!\cdots\!93}{27\!\cdots\!52}a^{5}+\frac{15\!\cdots\!23}{18\!\cdots\!84}a^{4}+\frac{59\!\cdots\!12}{23\!\cdots\!21}a^{3}+\frac{57\!\cdots\!03}{10\!\cdots\!02}a^{2}+\frac{52\!\cdots\!80}{59\!\cdots\!39}a+\frac{73\!\cdots\!83}{31\!\cdots\!81}$, $\frac{51\!\cdots\!93}{46\!\cdots\!33}a^{8}+\frac{28\!\cdots\!23}{55\!\cdots\!96}a^{7}-\frac{60\!\cdots\!90}{46\!\cdots\!33}a^{6}-\frac{31\!\cdots\!67}{30\!\cdots\!28}a^{5}+\frac{24\!\cdots\!91}{42\!\cdots\!04}a^{4}-\frac{15\!\cdots\!56}{77\!\cdots\!07}a^{3}+\frac{10\!\cdots\!55}{59\!\cdots\!39}a^{2}+\frac{30\!\cdots\!83}{17\!\cdots\!17}a+\frac{79\!\cdots\!57}{31\!\cdots\!81}$, $\frac{55\!\cdots\!66}{41\!\cdots\!97}a^{8}-\frac{16\!\cdots\!21}{18\!\cdots\!32}a^{7}-\frac{39\!\cdots\!41}{13\!\cdots\!99}a^{6}+\frac{37\!\cdots\!49}{27\!\cdots\!52}a^{5}+\frac{65\!\cdots\!59}{18\!\cdots\!84}a^{4}-\frac{86\!\cdots\!41}{46\!\cdots\!42}a^{3}-\frac{16\!\cdots\!01}{10\!\cdots\!02}a^{2}+\frac{37\!\cdots\!22}{59\!\cdots\!39}a+\frac{73\!\cdots\!04}{31\!\cdots\!81}$, $\frac{13\!\cdots\!69}{16\!\cdots\!88}a^{8}+\frac{96\!\cdots\!95}{13\!\cdots\!99}a^{7}-\frac{48\!\cdots\!83}{55\!\cdots\!96}a^{6}-\frac{30\!\cdots\!25}{27\!\cdots\!52}a^{5}+\frac{22\!\cdots\!50}{10\!\cdots\!01}a^{4}+\frac{29\!\cdots\!19}{46\!\cdots\!42}a^{3}+\frac{30\!\cdots\!53}{10\!\cdots\!02}a^{2}+\frac{34\!\cdots\!31}{17\!\cdots\!17}a+\frac{15\!\cdots\!84}{31\!\cdots\!81}$, $\frac{39\!\cdots\!81}{20\!\cdots\!61}a^{8}+\frac{35\!\cdots\!50}{20\!\cdots\!61}a^{7}-\frac{40\!\cdots\!48}{20\!\cdots\!61}a^{6}-\frac{91\!\cdots\!51}{33\!\cdots\!19}a^{5}+\frac{89\!\cdots\!79}{19\!\cdots\!07}a^{4}+\frac{49\!\cdots\!77}{33\!\cdots\!19}a^{3}+\frac{11\!\cdots\!66}{25\!\cdots\!63}a^{2}+\frac{12\!\cdots\!58}{25\!\cdots\!63}a+\frac{16\!\cdots\!63}{13\!\cdots\!77}$ (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: | \( 1296407.1079993355 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1296407.1079993355 \cdot 75759138}{2\cdot\sqrt{17971044653397089488481063155542528}}\cr\approx \mathstrut & 0.726924594453653 \end{aligned}\] (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
3.3.2934497241.1, 3.1.11112.1 |
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: | 6.0.6124062548879080540420608.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 | $x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
\(13\) | 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(463\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $6$ | $6$ | $1$ | $5$ |