Normalized defining polynomial
\( x^{18} - 3 x^{17} - 6 x^{16} + 27 x^{15} + 18 x^{14} - 129 x^{13} - 3 x^{12} + 354 x^{11} - 162 x^{10} - 464 x^{9} + 426 x^{8} + 150 x^{7} - 390 x^{6} + 252 x^{5} + 84 x^{4} - 216 x^{3} + \cdots + 4 \)
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: | \(-41451359947637504606208\) \(\medspace = -\,2^{26}\cdot 3^{31}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.05\) | 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: | $2^{11/6}3^{97/54}\approx 25.64126624149289$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{14}a^{15}-\frac{1}{14}a^{14}+\frac{1}{7}a^{13}-\frac{1}{7}a^{12}+\frac{1}{14}a^{11}+\frac{3}{14}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{1}{14}a^{7}+\frac{3}{14}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{3}+\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{238}a^{16}-\frac{3}{119}a^{15}-\frac{3}{17}a^{14}-\frac{33}{238}a^{13}+\frac{39}{238}a^{12}-\frac{79}{238}a^{11}-\frac{19}{238}a^{10}-\frac{117}{238}a^{9}+\frac{3}{17}a^{8}-\frac{43}{119}a^{7}+\frac{99}{238}a^{6}+\frac{30}{119}a^{5}-\frac{2}{7}a^{4}+\frac{2}{119}a^{3}-\frac{32}{119}a^{2}-\frac{5}{17}a-\frac{5}{119}$, $\frac{1}{9685173675758}a^{17}+\frac{2093318618}{4842586837879}a^{16}-\frac{155228015973}{9685173675758}a^{15}-\frac{2383119769407}{9685173675758}a^{14}+\frac{123474857847}{1383596239394}a^{13}+\frac{759973958645}{4842586837879}a^{12}+\frac{4182205203745}{9685173675758}a^{11}-\frac{2489213685323}{9685173675758}a^{10}+\frac{4676277163443}{9685173675758}a^{9}+\frac{1132457339017}{4842586837879}a^{8}-\frac{2842683667697}{9685173675758}a^{7}-\frac{1038359220419}{9685173675758}a^{6}+\frac{1028802196747}{4842586837879}a^{5}-\frac{1914193135891}{4842586837879}a^{4}+\frac{37584832387}{284858049287}a^{3}+\frac{84528298685}{4842586837879}a^{2}-\frac{1525991999611}{4842586837879}a-\frac{667678476036}{4842586837879}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -\frac{112159473}{390169346} a^{17} + \frac{298357011}{390169346} a^{16} + \frac{773170831}{390169346} a^{15} - \frac{1383174396}{195084673} a^{14} - \frac{2935100103}{390169346} a^{13} + \frac{960745367}{27869239} a^{12} + \frac{2383850487}{195084673} a^{11} - \frac{18962744301}{195084673} a^{10} + \frac{2896017147}{195084673} a^{9} + \frac{53298799815}{390169346} a^{8} - \frac{2183409615}{27869239} a^{7} - \frac{25591216657}{390169346} a^{6} + \frac{2522530224}{27869239} a^{5} - \frac{8851488483}{195084673} a^{4} - \frac{426128644}{11475569} a^{3} + \frac{1379281755}{27869239} a^{2} - \frac{4665195144}{195084673} a + \frac{881662172}{195084673} \) (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: | $\frac{4983793052212}{4842586837879}a^{17}-\frac{26519556670727}{9685173675758}a^{16}-\frac{2030235166020}{284858049287}a^{15}+\frac{246364831492071}{9685173675758}a^{14}+\frac{264312283344625}{9685173675758}a^{13}-\frac{12\!\cdots\!55}{9685173675758}a^{12}-\frac{63126314585035}{1383596239394}a^{11}+\frac{243272905061920}{691798119697}a^{10}-\frac{456419247702139}{9685173675758}a^{9}-\frac{24\!\cdots\!46}{4842586837879}a^{8}+\frac{13\!\cdots\!02}{4842586837879}a^{7}+\frac{12\!\cdots\!90}{4842586837879}a^{6}-\frac{15\!\cdots\!60}{4842586837879}a^{5}+\frac{706583435198161}{4842586837879}a^{4}+\frac{689541134467820}{4842586837879}a^{3}-\frac{854821036261648}{4842586837879}a^{2}+\frac{350920037376086}{4842586837879}a-\frac{6204286001285}{691798119697}$, $\frac{5882778895471}{9685173675758}a^{17}-\frac{15978020207809}{9685173675758}a^{16}-\frac{39949922142661}{9685173675758}a^{15}+\frac{147770957069735}{9685173675758}a^{14}+\frac{148742927981917}{9685173675758}a^{13}-\frac{359706250761495}{4842586837879}a^{12}-\frac{112851113703589}{4842586837879}a^{11}+\frac{20\!\cdots\!79}{9685173675758}a^{10}-\frac{26247862224754}{691798119697}a^{9}-\frac{28\!\cdots\!97}{9685173675758}a^{8}+\frac{49572953779404}{284858049287}a^{7}+\frac{710871837864061}{4842586837879}a^{6}-\frac{952443190982774}{4842586837879}a^{5}+\frac{64892921704763}{691798119697}a^{4}+\frac{387522565607621}{4842586837879}a^{3}-\frac{534351463632341}{4842586837879}a^{2}+\frac{222078265888106}{4842586837879}a-\frac{35491594119355}{4842586837879}$, $\frac{2251776782067}{9685173675758}a^{17}-\frac{241253965259}{569716098574}a^{16}-\frac{20032363977735}{9685173675758}a^{15}+\frac{20502707151028}{4842586837879}a^{14}+\frac{102159644544947}{9685173675758}a^{13}-\frac{206858054500247}{9685173675758}a^{12}-\frac{18221747245581}{569716098574}a^{11}+\frac{306619885436719}{4842586837879}a^{10}+\frac{71963959466041}{1383596239394}a^{9}-\frac{973448776888475}{9685173675758}a^{8}-\frac{311210934593981}{9685173675758}a^{7}+\frac{369989891617550}{4842586837879}a^{6}-\frac{75003332451965}{4842586837879}a^{5}-\frac{32471394472884}{4842586837879}a^{4}+\frac{214007562051221}{4842586837879}a^{3}-\frac{44053426264807}{4842586837879}a^{2}-\frac{30031835203190}{4842586837879}a+\frac{12511127241345}{4842586837879}$, $\frac{1853731809712}{4842586837879}a^{17}-\frac{675553079421}{691798119697}a^{16}-\frac{26810000805757}{9685173675758}a^{15}+\frac{88709006141811}{9685173675758}a^{14}+\frac{54671373994526}{4842586837879}a^{13}-\frac{435826209654265}{9685173675758}a^{12}-\frac{110092733283444}{4842586837879}a^{11}+\frac{12\!\cdots\!87}{9685173675758}a^{10}-\frac{3170022593337}{9685173675758}a^{9}-\frac{912644987744447}{4842586837879}a^{8}+\frac{356449119797224}{4842586837879}a^{7}+\frac{521703114579034}{4842586837879}a^{6}-\frac{28913871019556}{284858049287}a^{5}+\frac{188010193687300}{4842586837879}a^{4}+\frac{274512535790600}{4842586837879}a^{3}-\frac{285171945436519}{4842586837879}a^{2}+\frac{83867904499065}{4842586837879}a-\frac{512749223373}{284858049287}$, $\frac{307229777049}{4842586837879}a^{17}-\frac{4087141389973}{9685173675758}a^{16}+\frac{795553273447}{4842586837879}a^{15}+\frac{33686722803765}{9685173675758}a^{14}-\frac{5563502468331}{1383596239394}a^{13}-\frac{153729314790101}{9685173675758}a^{12}+\frac{244182584304157}{9685173675758}a^{11}+\frac{11504894481307}{284858049287}a^{10}-\frac{805571340878527}{9685173675758}a^{9}-\frac{198660935488426}{4842586837879}a^{8}+\frac{650864893334658}{4842586837879}a^{7}-\frac{87838668240164}{4842586837879}a^{6}-\frac{430759644264860}{4842586837879}a^{5}+\frac{46649967639232}{691798119697}a^{4}-\frac{58550613258457}{4842586837879}a^{3}-\frac{244462729209141}{4842586837879}a^{2}+\frac{175539419014900}{4842586837879}a-\frac{50591779160635}{4842586837879}$, $\frac{9715294164679}{9685173675758}a^{17}-\frac{1742800503214}{691798119697}a^{16}-\frac{35502964957860}{4842586837879}a^{15}+\frac{32808571736401}{1383596239394}a^{14}+\frac{292335197896933}{9685173675758}a^{13}-\frac{563942630002903}{4842586837879}a^{12}-\frac{301106933686229}{4842586837879}a^{11}+\frac{32\!\cdots\!85}{9685173675758}a^{10}+\frac{49090741229753}{9685173675758}a^{9}-\frac{23\!\cdots\!51}{4842586837879}a^{8}+\frac{908240096230406}{4842586837879}a^{7}+\frac{26\!\cdots\!51}{9685173675758}a^{6}-\frac{12\!\cdots\!81}{4842586837879}a^{5}+\frac{513968294976078}{4842586837879}a^{4}+\frac{715080720125038}{4842586837879}a^{3}-\frac{723387451197230}{4842586837879}a^{2}+\frac{241101652563476}{4842586837879}a-\frac{4339968489950}{691798119697}$, $\frac{1079482508886}{4842586837879}a^{17}-\frac{420547107586}{691798119697}a^{16}-\frac{14586904117017}{9685173675758}a^{15}+\frac{54448611167815}{9685173675758}a^{14}+\frac{26828116886674}{4842586837879}a^{13}-\frac{132374547755475}{4842586837879}a^{12}-\frac{11070193628099}{1383596239394}a^{11}+\frac{746130571046923}{9685173675758}a^{10}-\frac{10933504726333}{691798119697}a^{9}-\frac{74762335054469}{691798119697}a^{8}+\frac{652820861823491}{9685173675758}a^{7}+\frac{28760995637253}{569716098574}a^{6}-\frac{361617821165679}{4842586837879}a^{5}+\frac{191364401688764}{4842586837879}a^{4}+\frac{135187896259190}{4842586837879}a^{3}-\frac{208995524799245}{4842586837879}a^{2}+\frac{95208336294538}{4842586837879}a-\frac{15644641880655}{4842586837879}$, $\frac{1079482508886}{4842586837879}a^{17}-\frac{420547107586}{691798119697}a^{16}-\frac{14586904117017}{9685173675758}a^{15}+\frac{54448611167815}{9685173675758}a^{14}+\frac{26828116886674}{4842586837879}a^{13}-\frac{132374547755475}{4842586837879}a^{12}-\frac{11070193628099}{1383596239394}a^{11}+\frac{746130571046923}{9685173675758}a^{10}-\frac{10933504726333}{691798119697}a^{9}-\frac{74762335054469}{691798119697}a^{8}+\frac{652820861823491}{9685173675758}a^{7}+\frac{28760995637253}{569716098574}a^{6}-\frac{361617821165679}{4842586837879}a^{5}+\frac{191364401688764}{4842586837879}a^{4}+\frac{135187896259190}{4842586837879}a^{3}-\frac{208995524799245}{4842586837879}a^{2}+\frac{95208336294538}{4842586837879}a-\frac{10802055042776}{4842586837879}$ | 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: | \( 42016.82501482997 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 42016.82501482997 \cdot 1}{6\cdot\sqrt{41451359947637504606208}}\cr\approx \mathstrut & 0.524954188320811 \end{aligned}\]
Galois group
$C_3\times S_3^2$ (as 18T46):
A solvable group of order 108 |
The 27 conjugacy class representatives for $C_3\times S_3^2$ |
Character table for $C_3\times S_3^2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1, 6.0.314928.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.1624959306694656.38 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
2.12.22.27 | $x^{12} + 8 x^{9} + 16 x^{8} + 4 x^{7} + 2 x^{6} + 40 x^{5} + 104 x^{4} - 48 x^{3} - 12 x^{2} + 56 x + 196$ | $6$ | $2$ | $22$ | $C_6\times S_3$ | $[3]_{3}^{6}$ | |
\(3\) | Deg $18$ | $18$ | $1$ | $31$ |