Normalized defining polynomial
\( x^{9} - 3 x^{8} - 98565 x^{7} - 6249892 x^{6} + 3254568816 x^{5} + 420957601977 x^{4} + \cdots - 99\!\cdots\!79 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-362790052878945346803558493577885799936\) \(\medspace = -\,2^{9}\cdot 3^{12}\cdot 11^{3}\cdot 19^{6}\cdot 577^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19\,248.91\) | 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: | $2^{3/2}3^{4/3}11^{1/2}19^{2/3}577^{5/6}\approx 57794.3327376087$ | ||
Ramified primes: | \(2\), \(3\), \(11\), \(19\), \(577\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-12694}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{41}a^{5}-\frac{16}{41}a^{4}+\frac{5}{41}a^{3}-\frac{2}{41}a^{2}-\frac{16}{41}a$, $\frac{1}{2129130}a^{6}-\frac{1337}{354855}a^{5}-\frac{18263}{141942}a^{4}-\frac{230551}{1064565}a^{3}+\frac{89791}{354855}a^{2}+\frac{12533}{118285}a+\frac{2237}{17310}$, $\frac{1}{2129130}a^{7}-\frac{8503}{709710}a^{5}+\frac{178364}{1064565}a^{4}-\frac{41436}{118285}a^{3}+\frac{32622}{118285}a^{2}-\frac{252317}{709710}a-\frac{876}{2885}$, $\frac{1}{22\!\cdots\!30}a^{8}+\frac{20\!\cdots\!03}{22\!\cdots\!30}a^{7}+\frac{18\!\cdots\!81}{22\!\cdots\!30}a^{6}-\frac{22\!\cdots\!09}{22\!\cdots\!30}a^{5}-\frac{29\!\cdots\!57}{11\!\cdots\!15}a^{4}+\frac{34\!\cdots\!31}{11\!\cdots\!15}a^{3}+\frac{65\!\cdots\!23}{25\!\cdots\!70}a^{2}+\frac{11\!\cdots\!93}{75\!\cdots\!10}a+\frac{14\!\cdots\!03}{24\!\cdots\!65}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{6832098}$, which has order $3320399628$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{17\!\cdots\!24}{12\!\cdots\!35}a^{8}-\frac{12\!\cdots\!89}{75\!\cdots\!10}a^{7}-\frac{43\!\cdots\!34}{37\!\cdots\!05}a^{6}+\frac{42\!\cdots\!13}{84\!\cdots\!90}a^{5}+\frac{14\!\cdots\!93}{37\!\cdots\!05}a^{4}+\frac{89\!\cdots\!55}{75\!\cdots\!61}a^{3}-\frac{11\!\cdots\!10}{25\!\cdots\!87}a^{2}-\frac{10\!\cdots\!47}{25\!\cdots\!70}a-\frac{18\!\cdots\!77}{16\!\cdots\!11}$, $\frac{78\!\cdots\!37}{75\!\cdots\!10}a^{8}+\frac{10\!\cdots\!99}{12\!\cdots\!35}a^{7}+\frac{72\!\cdots\!09}{75\!\cdots\!10}a^{6}-\frac{37\!\cdots\!27}{37\!\cdots\!05}a^{5}-\frac{41\!\cdots\!27}{12\!\cdots\!35}a^{4}-\frac{67\!\cdots\!08}{37\!\cdots\!05}a^{3}+\frac{92\!\cdots\!19}{25\!\cdots\!70}a^{2}+\frac{18\!\cdots\!32}{42\!\cdots\!45}a+\frac{10\!\cdots\!32}{83\!\cdots\!55}$, $\frac{46\!\cdots\!44}{12\!\cdots\!35}a^{8}-\frac{62\!\cdots\!83}{15\!\cdots\!22}a^{7}-\frac{26\!\cdots\!33}{75\!\cdots\!10}a^{6}+\frac{93\!\cdots\!89}{84\!\cdots\!90}a^{5}+\frac{94\!\cdots\!03}{75\!\cdots\!10}a^{4}+\frac{19\!\cdots\!94}{37\!\cdots\!05}a^{3}-\frac{19\!\cdots\!56}{12\!\cdots\!35}a^{2}-\frac{81\!\cdots\!01}{50\!\cdots\!74}a-\frac{76\!\cdots\!67}{16\!\cdots\!10}$, $\frac{74\!\cdots\!89}{75\!\cdots\!10}a^{8}-\frac{31\!\cdots\!79}{25\!\cdots\!70}a^{7}-\frac{30\!\cdots\!67}{37\!\cdots\!05}a^{6}+\frac{30\!\cdots\!51}{75\!\cdots\!10}a^{5}+\frac{68\!\cdots\!79}{25\!\cdots\!70}a^{4}+\frac{28\!\cdots\!88}{37\!\cdots\!05}a^{3}-\frac{79\!\cdots\!13}{25\!\cdots\!70}a^{2}-\frac{24\!\cdots\!03}{84\!\cdots\!90}a-\frac{13\!\cdots\!77}{16\!\cdots\!10}$, $\frac{13\!\cdots\!08}{37\!\cdots\!05}a^{8}-\frac{10\!\cdots\!28}{37\!\cdots\!05}a^{7}-\frac{12\!\cdots\!82}{37\!\cdots\!05}a^{6}+\frac{14\!\cdots\!46}{37\!\cdots\!05}a^{5}+\frac{42\!\cdots\!87}{37\!\cdots\!05}a^{4}+\frac{22\!\cdots\!62}{37\!\cdots\!05}a^{3}-\frac{53\!\cdots\!89}{42\!\cdots\!45}a^{2}-\frac{18\!\cdots\!57}{12\!\cdots\!35}a-\frac{35\!\cdots\!76}{83\!\cdots\!55}$ (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: | \( 5182831.720989658 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 5182831.720989658 \cdot 3320399628}{2\cdot\sqrt{362790052878945346803558493577885799936}}\cr\approx \mathstrut & 0.896456820171234 \end{aligned}\] (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
3.3.9735176889.1, 3.1.50776.1 |
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: | 6.0.37265892239602771002465191424.1 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.1.0.1}{1} }^{9}$ | ${\href{/padicField/41.1.0.1}{1} }^{9}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(3\) | 3.3.4.1 | $x^{3} + 6 x^{2} + 21$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.6.8.6 | $x^{6} + 18 x^{5} + 114 x^{4} + 362 x^{3} + 894 x^{2} + 960 x + 557$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
\(11\) | 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(19\) | 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(577\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $6$ | $6$ | $1$ | $5$ |