Normalized defining polynomial
\( x^{18} - x^{17} - x^{16} + x^{15} + 3 x^{14} + 2 x^{13} - 5 x^{12} + 4 x^{11} + 5 x^{10} + 5 x^{9} + \cdots + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-14929107797061341783\) \(\medspace = -\,23^{9}\cdot 2879^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.62\) | 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: | $23^{1/2}2879^{1/2}\approx 257.3266406729004$ | ||
Ramified primes: | \(23\), \(2879\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{33}a^{16}+\frac{4}{33}a^{15}-\frac{2}{33}a^{14}-\frac{5}{33}a^{13}-\frac{2}{33}a^{12}-\frac{13}{33}a^{11}+\frac{5}{33}a^{10}+\frac{16}{33}a^{9}+\frac{2}{33}a^{8}+\frac{3}{11}a^{7}-\frac{13}{33}a^{6}+\frac{2}{11}a^{5}+\frac{1}{11}a^{4}+\frac{5}{33}a^{3}+\frac{1}{3}a^{2}-\frac{16}{33}a+\frac{10}{33}$, $\frac{1}{1693593}a^{17}-\frac{47}{17107}a^{16}+\frac{142256}{1693593}a^{15}+\frac{23852}{188177}a^{14}+\frac{40816}{564531}a^{13}-\frac{324241}{1693593}a^{12}+\frac{4993}{188177}a^{11}-\frac{67808}{1693593}a^{10}+\frac{247483}{564531}a^{9}+\frac{800372}{1693593}a^{8}+\frac{777827}{1693593}a^{7}+\frac{295144}{1693593}a^{6}+\frac{10718}{564531}a^{5}+\frac{276907}{1693593}a^{4}-\frac{210037}{564531}a^{3}+\frac{539347}{1693593}a^{2}-\frac{277003}{564531}a+\frac{202184}{1693593}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{631658}{1693593}a^{17}-\frac{223166}{564531}a^{16}-\frac{67921}{153963}a^{15}+\frac{275650}{564531}a^{14}+\frac{221895}{188177}a^{13}+\frac{1193518}{1693593}a^{12}-\frac{1285831}{564531}a^{11}+\frac{2016035}{1693593}a^{10}+\frac{1364024}{564531}a^{9}+\frac{3081271}{1693593}a^{8}+\frac{3005383}{1693593}a^{7}-\frac{4703047}{1693593}a^{6}+\frac{1486396}{564531}a^{5}-\frac{1351147}{1693593}a^{4}+\frac{219364}{188177}a^{3}+\frac{1373039}{1693593}a^{2}-\frac{320192}{564531}a-\frac{500255}{1693593}$, $\frac{447739}{1693593}a^{17}-\frac{87166}{564531}a^{16}-\frac{38246}{153963}a^{15}-\frac{1797}{188177}a^{14}+\frac{379381}{564531}a^{13}+\frac{1811627}{1693593}a^{12}-\frac{287939}{564531}a^{11}+\frac{1063525}{1693593}a^{10}+\frac{81161}{188177}a^{9}+\frac{3539024}{1693593}a^{8}+\frac{5540957}{1693593}a^{7}-\frac{92546}{1693593}a^{6}+\frac{208938}{188177}a^{5}-\frac{4340627}{1693593}a^{4}+\frac{742934}{564531}a^{3}+\frac{83218}{1693593}a^{2}+\frac{618248}{564531}a+\frac{909323}{1693593}$, $\frac{344261}{1693593}a^{17}-\frac{8242}{51321}a^{16}-\frac{435965}{1693593}a^{15}+\frac{21800}{188177}a^{14}+\frac{368563}{564531}a^{13}+\frac{1183729}{1693593}a^{12}-\frac{493343}{564531}a^{11}+\frac{271493}{1693593}a^{10}+\frac{479251}{564531}a^{9}+\frac{2832736}{1693593}a^{8}+\frac{3205210}{1693593}a^{7}-\frac{2237044}{1693593}a^{6}+\frac{391136}{564531}a^{5}-\frac{682057}{1693593}a^{4}+\frac{197885}{188177}a^{3}+\frac{1327136}{1693593}a^{2}+\frac{375799}{564531}a+\frac{780910}{1693593}$, $\frac{18500}{188177}a^{17}-\frac{49397}{188177}a^{16}-\frac{100175}{564531}a^{15}+\frac{194669}{564531}a^{14}+\frac{279320}{564531}a^{13}-\frac{214274}{564531}a^{12}-\frac{288091}{188177}a^{11}+\frac{144679}{564531}a^{10}+\frac{61986}{188177}a^{9}-\frac{133171}{188177}a^{8}-\frac{868708}{564531}a^{7}-\frac{1650196}{564531}a^{6}-\frac{75005}{51321}a^{5}-\frac{424860}{188177}a^{4}+\frac{166390}{564531}a^{3}+\frac{22252}{188177}a^{2}-\frac{82718}{564531}a-\frac{212620}{188177}$, $\frac{500255}{1693593}a^{17}+\frac{43801}{564531}a^{16}-\frac{1169753}{1693593}a^{15}-\frac{82292}{564531}a^{14}+\frac{258635}{188177}a^{13}+\frac{2997565}{1693593}a^{12}-\frac{39629}{51321}a^{11}-\frac{1856473}{1693593}a^{10}+\frac{1505770}{564531}a^{9}+\frac{6593347}{1693593}a^{8}+\frac{6082801}{1693593}a^{7}+\frac{1004363}{1693593}a^{6}-\frac{233669}{564531}a^{5}+\frac{1957913}{1693593}a^{4}+\frac{127792}{188177}a^{3}+\frac{2474531}{1693593}a^{2}+\frac{791183}{564531}a+\frac{232762}{1693593}$, $\frac{12775}{153963}a^{17}-\frac{113737}{564531}a^{16}-\frac{339446}{1693593}a^{15}+\frac{117584}{564531}a^{14}+\frac{375580}{564531}a^{13}-\frac{258947}{1693593}a^{12}-\frac{1016153}{564531}a^{11}-\frac{462340}{1693593}a^{10}+\frac{591419}{564531}a^{9}+\frac{918133}{1693593}a^{8}-\frac{3894194}{1693593}a^{7}-\frac{6781243}{1693593}a^{6}-\frac{432326}{564531}a^{5}-\frac{816976}{1693593}a^{4}+\frac{711770}{564531}a^{3}-\frac{148214}{153963}a^{2}-\frac{151100}{564531}a-\frac{168815}{1693593}$, $\frac{493513}{1693593}a^{17}-\frac{80851}{188177}a^{16}-\frac{17978}{153963}a^{15}+\frac{65125}{188177}a^{14}+\frac{418388}{564531}a^{13}+\frac{446342}{1693593}a^{12}-\frac{903605}{564531}a^{11}+\frac{2859127}{1693593}a^{10}+\frac{326246}{564531}a^{9}+\frac{2218262}{1693593}a^{8}+\frac{2052551}{1693593}a^{7}-\frac{3862847}{1693593}a^{6}+\frac{577616}{188177}a^{5}-\frac{4408868}{1693593}a^{4}+\frac{488144}{188177}a^{3}-\frac{1045958}{1693593}a^{2}+\frac{258073}{188177}a-\frac{627142}{1693593}$, $\frac{962}{564531}a^{17}+\frac{211132}{564531}a^{16}-\frac{211012}{564531}a^{15}-\frac{87231}{188177}a^{14}+\frac{89962}{188177}a^{13}+\frac{676369}{564531}a^{12}+\frac{359252}{564531}a^{11}-\frac{1148474}{564531}a^{10}+\frac{257132}{188177}a^{9}+\frac{1220605}{564531}a^{8}+\frac{288248}{188177}a^{7}+\frac{440296}{188177}a^{6}-\frac{1160584}{564531}a^{5}+\frac{1191820}{564531}a^{4}-\frac{26876}{17107}a^{3}+\frac{1176887}{564531}a^{2}+\frac{218950}{564531}a+\frac{29041}{51321}$ | 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: | \( 92.6925275389 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 92.6925275389 \cdot 1}{2\cdot\sqrt{14929107797061341783}}\cr\approx \mathstrut & 0.183069762137 \end{aligned}\]
Galois group
$S_3\wr S_3$ (as 18T319):
A solvable group of order 1296 |
The 22 conjugacy class representatives for $S_3\wr S_3$ |
Character table for $S_3\wr S_3$ |
Intermediate fields
\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.35028793.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 9.1.35028793.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{6}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(2879\) | $\Q_{2879}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2879}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2879}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2879}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2879}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2879}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |