Normalized defining polynomial
\( x^{18} - x^{17} + 4 x^{16} - x^{15} + 12 x^{14} - 9 x^{13} + 12 x^{12} - 18 x^{11} + 26 x^{10} - 12 x^{9} - 4 x^{8} - 5 x^{7} + 31 x^{6} - 17 x^{5} - 3 x^{4} + 5 x^{2} - 3 x + 1 \)
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: | \(-43564677551979246963\) \(\medspace = -\,3^{9}\cdot 19^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.33\) | 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: | $3^{1/2}19^{2/3}\approx 12.332838034173273$ | ||
Ramified primes: | \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{78}a^{15}-\frac{3}{26}a^{14}+\frac{3}{26}a^{13}-\frac{1}{78}a^{12}+\frac{6}{13}a^{11}-\frac{17}{78}a^{10}+\frac{35}{78}a^{9}+\frac{1}{26}a^{8}+\frac{1}{39}a^{7}+\frac{5}{39}a^{6}+\frac{35}{78}a^{5}+\frac{11}{78}a^{4}-\frac{8}{39}a^{3}-\frac{23}{78}a^{2}+\frac{7}{39}a-\frac{7}{78}$, $\frac{1}{858}a^{16}-\frac{2}{429}a^{15}-\frac{19}{143}a^{14}-\frac{4}{429}a^{13}+\frac{83}{858}a^{12}+\frac{397}{858}a^{11}+\frac{9}{143}a^{10}-\frac{70}{143}a^{9}-\frac{133}{286}a^{8}-\frac{185}{429}a^{7}-\frac{23}{78}a^{6}-\frac{37}{429}a^{5}-\frac{1}{2}a^{4}-\frac{11}{26}a^{3}+\frac{107}{858}a^{2}-\frac{145}{858}a-\frac{347}{858}$, $\frac{1}{1570998}a^{17}-\frac{7}{40282}a^{16}-\frac{8135}{1570998}a^{15}-\frac{156595}{1570998}a^{14}-\frac{2286}{20141}a^{13}+\frac{256579}{1570998}a^{12}-\frac{616127}{1570998}a^{11}+\frac{11575}{1570998}a^{10}+\frac{289771}{785499}a^{9}+\frac{177115}{785499}a^{8}+\frac{56867}{523666}a^{7}-\frac{122825}{523666}a^{6}-\frac{29609}{261833}a^{5}-\frac{69749}{142818}a^{4}-\frac{177217}{785499}a^{3}-\frac{170047}{1570998}a^{2}-\frac{254560}{785499}a+\frac{9201}{261833}$
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: | \( \frac{16945}{142818} a^{17} - \frac{23327}{71409} a^{16} + \frac{36250}{71409} a^{15} - \frac{21461}{23803} a^{14} + \frac{47967}{47606} a^{13} - \frac{549617}{142818} a^{12} + \frac{26899}{23803} a^{11} - \frac{324175}{71409} a^{10} + \frac{753937}{142818} a^{9} - \frac{346456}{71409} a^{8} - \frac{106321}{142818} a^{7} - \frac{17395}{71409} a^{6} + \frac{814865}{142818} a^{5} - \frac{345281}{47606} a^{4} - \frac{194975}{142818} a^{3} - \frac{34673}{142818} a^{2} + \frac{95847}{47606} a - \frac{6029}{23803} \) (order $6$) | 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: | $\frac{131732}{785499}a^{17}-\frac{294956}{785499}a^{16}+\frac{7037}{10986}a^{15}-\frac{457835}{523666}a^{14}+\frac{1958383}{1570998}a^{13}-\frac{2172241}{523666}a^{12}+\frac{497582}{785499}a^{11}-\frac{7582787}{1570998}a^{10}+\frac{7294097}{1570998}a^{9}-\frac{6308743}{1570998}a^{8}-\frac{2785624}{785499}a^{7}+\frac{116374}{60423}a^{6}+\frac{3492409}{523666}a^{5}-\frac{996653}{142818}a^{4}-\frac{1991395}{785499}a^{3}+\frac{834461}{523666}a^{2}+\frac{175514}{261833}a-\frac{49527}{523666}$, $\frac{560}{785499}a^{17}+\frac{9323}{142818}a^{16}+\frac{30983}{1570998}a^{15}+\frac{13417}{40282}a^{14}+\frac{7991}{40282}a^{13}+\frac{340625}{261833}a^{12}+\frac{819835}{1570998}a^{11}+\frac{3169981}{1570998}a^{10}-\frac{84949}{142818}a^{9}+\frac{1633343}{785499}a^{8}-\frac{215290}{261833}a^{7}+\frac{945657}{523666}a^{6}-\frac{779853}{523666}a^{5}+\frac{1770}{1831}a^{4}-\frac{168239}{1570998}a^{3}+\frac{1170337}{785499}a^{2}-\frac{2027635}{1570998}a-\frac{138400}{785499}$, $\frac{138400}{785499}a^{17}-\frac{137840}{785499}a^{16}+\frac{403251}{523666}a^{15}-\frac{573}{3662}a^{14}+\frac{116511}{47606}a^{13}-\frac{66047}{47606}a^{12}+\frac{894225}{261833}a^{11}-\frac{378415}{142818}a^{10}+\frac{10366781}{1570998}a^{9}-\frac{4256039}{1570998}a^{8}+\frac{1079743}{785499}a^{7}-\frac{1337870}{785499}a^{6}+\frac{11417771}{1570998}a^{5}-\frac{640469}{142818}a^{4}+\frac{114710}{261833}a^{3}-\frac{168239}{1570998}a^{2}+\frac{620779}{261833}a-\frac{1287037}{1570998}$, $\frac{130265}{523666}a^{17}+\frac{74992}{785499}a^{16}+\frac{56861}{71409}a^{15}+\frac{813652}{785499}a^{14}+\frac{1655091}{523666}a^{13}+\frac{2992549}{1570998}a^{12}+\frac{414400}{261833}a^{11}-\frac{339720}{261833}a^{10}+\frac{2212201}{1570998}a^{9}+\frac{2475851}{785499}a^{8}-\frac{1181269}{523666}a^{7}-\frac{1041855}{261833}a^{6}+\frac{2860431}{523666}a^{5}+\frac{602461}{142818}a^{4}-\frac{788313}{523666}a^{3}-\frac{814745}{523666}a^{2}-\frac{774733}{1570998}a-\frac{9719}{785499}$, $\frac{71515}{785499}a^{17}+\frac{99232}{785499}a^{16}+\frac{72660}{261833}a^{15}+\frac{630640}{785499}a^{14}+\frac{318849}{261833}a^{13}+\frac{1775770}{785499}a^{12}+\frac{289775}{785499}a^{11}+\frac{1133569}{785499}a^{10}-\frac{534795}{261833}a^{9}+\frac{831295}{261833}a^{8}-\frac{3040541}{785499}a^{7}-\frac{213685}{785499}a^{6}-\frac{1054145}{785499}a^{5}+\frac{221735}{71409}a^{4}+\frac{100990}{785499}a^{3}+\frac{29090}{261833}a^{2}-\frac{1144201}{785499}a+\frac{15245}{261833}$, $\frac{11201}{60423}a^{17}-\frac{5165}{523666}a^{16}+\frac{750743}{785499}a^{15}+\frac{279358}{785499}a^{14}+\frac{936820}{261833}a^{13}+\frac{1476997}{1570998}a^{12}+\frac{9095929}{1570998}a^{11}-\frac{1193626}{785499}a^{10}+\frac{5317330}{785499}a^{9}-\frac{183259}{120846}a^{8}+\frac{99692}{20141}a^{7}-\frac{1048871}{523666}a^{6}+\frac{877815}{261833}a^{5}+\frac{100597}{142818}a^{4}+\frac{9580381}{1570998}a^{3}-\frac{2476423}{1570998}a^{2}+\frac{120113}{142818}a+\frac{380617}{523666}$, $\frac{219748}{785499}a^{17}-\frac{215661}{523666}a^{16}+\frac{931243}{785499}a^{15}-\frac{479137}{785499}a^{14}+\frac{838769}{261833}a^{13}-\frac{5565673}{1570998}a^{12}+\frac{6304385}{1570998}a^{11}-\frac{3367196}{785499}a^{10}+\frac{7130768}{785499}a^{9}-\frac{6556985}{1570998}a^{8}-\frac{637882}{261833}a^{7}+\frac{1114115}{523666}a^{6}+\frac{2802212}{261833}a^{5}-\frac{1263811}{142818}a^{4}-\frac{2495917}{1570998}a^{3}+\frac{4347193}{1570998}a^{2}+\frac{3277271}{1570998}a-\frac{718093}{523666}$, $\frac{58948}{785499}a^{17}-\frac{22651}{71409}a^{16}+\frac{135688}{261833}a^{15}-\frac{795848}{785499}a^{14}+\frac{268977}{261833}a^{13}-\frac{922538}{261833}a^{12}+\frac{710962}{261833}a^{11}-\frac{3087947}{785499}a^{10}+\frac{404140}{71409}a^{9}-\frac{5125552}{785499}a^{8}+\frac{962792}{785499}a^{7}+\frac{1224341}{785499}a^{6}+\frac{1735535}{785499}a^{5}-\frac{575387}{71409}a^{4}+\frac{1218814}{785499}a^{3}+\frac{184774}{60423}a^{2}-\frac{10273}{261833}a-\frac{1002175}{785499}$ | 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: | \( 527.323608131 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 527.323608131 \cdot 1}{6\cdot\sqrt{43564677551979246963}}\cr\approx \mathstrut & 0.203225185150 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 18T3):
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
\(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 3.3.361.1, 6.0.3518667.2, 6.0.9747.1 x2, 6.0.3518667.1, 9.3.1270238787.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 sibling: | 6.0.9747.1 |
Degree 9 sibling: | 9.3.1270238787.1 |
Minimal sibling: | 6.0.9747.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(19\) | 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 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
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 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |