Normalized defining polynomial
\( x^{18} - 2 x^{17} - 6 x^{16} + 22 x^{15} + 24 x^{14} - 200 x^{13} + 553 x^{12} - 586 x^{11} + \cdots + 8471 \)
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: | \(-502592611936843000000000000\) \(\medspace = -\,2^{12}\cdot 5^{12}\cdot 43^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(30.44\) | 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: | $2^{2/3}5^{2/3}43^{1/2}\approx 30.436933431502425$ | ||
Ramified primes: | \(2\), \(5\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-43}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}$, $\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{18}a^{8}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{7}{18}a^{4}-\frac{1}{2}a^{3}+\frac{1}{18}a^{2}-\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{18}a^{11}+\frac{1}{18}a^{9}+\frac{1}{18}a^{8}-\frac{1}{9}a^{7}-\frac{1}{6}a^{6}+\frac{1}{18}a^{5}-\frac{4}{9}a^{4}+\frac{1}{18}a^{3}+\frac{1}{6}a^{2}-\frac{1}{9}a+\frac{7}{18}$, $\frac{1}{378}a^{12}+\frac{5}{189}a^{11}-\frac{23}{378}a^{9}-\frac{1}{18}a^{8}+\frac{13}{378}a^{7}+\frac{2}{63}a^{6}+\frac{53}{189}a^{5}-\frac{17}{63}a^{4}+\frac{85}{378}a^{3}+\frac{5}{18}a^{2}+\frac{79}{378}a-\frac{85}{378}$, $\frac{1}{756}a^{13}+\frac{5}{756}a^{11}+\frac{19}{756}a^{10}+\frac{41}{756}a^{9}-\frac{29}{756}a^{8}+\frac{113}{756}a^{7}-\frac{5}{108}a^{6}-\frac{19}{108}a^{5}-\frac{155}{756}a^{4}-\frac{325}{756}a^{3}-\frac{47}{756}a^{2}+\frac{11}{54}a+\frac{283}{756}$, $\frac{1}{6804}a^{14}+\frac{1}{1701}a^{13}+\frac{1}{972}a^{12}+\frac{143}{6804}a^{11}-\frac{5}{252}a^{10}-\frac{79}{6804}a^{9}+\frac{181}{2268}a^{8}+\frac{779}{6804}a^{7}+\frac{253}{2268}a^{6}+\frac{2129}{6804}a^{5}+\frac{653}{2268}a^{4}-\frac{3109}{6804}a^{3}+\frac{800}{1701}a^{2}-\frac{305}{972}a-\frac{233}{3402}$, $\frac{1}{34020}a^{15}-\frac{1}{34020}a^{14}-\frac{1}{8505}a^{13}+\frac{1}{1890}a^{12}-\frac{193}{34020}a^{11}+\frac{389}{34020}a^{10}-\frac{2293}{34020}a^{9}+\frac{449}{34020}a^{8}-\frac{2911}{34020}a^{7}+\frac{1097}{34020}a^{6}-\frac{12619}{34020}a^{5}-\frac{5497}{34020}a^{4}-\frac{1832}{8505}a^{3}-\frac{316}{945}a^{2}-\frac{3419}{11340}a-\frac{15067}{34020}$, $\frac{1}{612360}a^{16}+\frac{1}{153090}a^{15}+\frac{41}{612360}a^{14}-\frac{1}{8505}a^{13}-\frac{113}{612360}a^{12}-\frac{19}{43740}a^{11}-\frac{2353}{102060}a^{10}+\frac{3121}{153090}a^{9}+\frac{148}{3645}a^{8}-\frac{8567}{153090}a^{7}-\frac{2203}{34020}a^{6}+\frac{149159}{306180}a^{5}+\frac{192737}{612360}a^{4}+\frac{9857}{51030}a^{3}+\frac{11029}{87480}a^{2}+\frac{18041}{76545}a+\frac{3649}{17496}$, $\frac{1}{48\!\cdots\!00}a^{17}+\frac{2388373513}{64\!\cdots\!44}a^{16}-\frac{4741090498601}{48\!\cdots\!00}a^{15}-\frac{273338690923}{97\!\cdots\!60}a^{14}-\frac{32599444926953}{69\!\cdots\!00}a^{13}+\frac{1831368972487}{25\!\cdots\!00}a^{12}+\frac{12\!\cdots\!48}{60\!\cdots\!25}a^{11}+\frac{401395420656971}{24\!\cdots\!00}a^{10}+\frac{44\!\cdots\!09}{12\!\cdots\!50}a^{9}+\frac{352287005819233}{12\!\cdots\!50}a^{8}-\frac{91\!\cdots\!03}{24\!\cdots\!00}a^{7}+\frac{20\!\cdots\!29}{17\!\cdots\!50}a^{6}-\frac{56\!\cdots\!01}{48\!\cdots\!00}a^{5}-\frac{24\!\cdots\!31}{69\!\cdots\!00}a^{4}+\frac{67\!\cdots\!09}{97\!\cdots\!60}a^{3}-\frac{16\!\cdots\!31}{16\!\cdots\!00}a^{2}-\frac{32\!\cdots\!91}{97\!\cdots\!60}a+\frac{44\!\cdots\!97}{48\!\cdots\!00}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$
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: | $\frac{9899839656773}{69\!\cdots\!00}a^{17}-\frac{2796864798857}{24\!\cdots\!90}a^{16}-\frac{482029660379411}{48\!\cdots\!00}a^{15}+\frac{10579674442409}{53\!\cdots\!20}a^{14}+\frac{407620447231487}{69\!\cdots\!00}a^{13}-\frac{52\!\cdots\!01}{24\!\cdots\!00}a^{12}+\frac{151533000347938}{28\!\cdots\!25}a^{11}-\frac{670811745342227}{34\!\cdots\!00}a^{10}+\frac{37\!\cdots\!51}{80\!\cdots\!00}a^{9}-\frac{25\!\cdots\!09}{24\!\cdots\!00}a^{8}+\frac{29\!\cdots\!93}{44\!\cdots\!50}a^{7}-\frac{12\!\cdots\!71}{17\!\cdots\!50}a^{6}+\frac{14\!\cdots\!29}{48\!\cdots\!00}a^{5}-\frac{25\!\cdots\!77}{80\!\cdots\!00}a^{4}+\frac{15\!\cdots\!61}{97\!\cdots\!60}a^{3}-\frac{40\!\cdots\!57}{12\!\cdots\!50}a^{2}+\frac{25\!\cdots\!33}{13\!\cdots\!80}a-\frac{45\!\cdots\!53}{38\!\cdots\!00}$, $\frac{996435568739}{20\!\cdots\!75}a^{17}-\frac{1891042156733}{97\!\cdots\!60}a^{16}-\frac{88522488245393}{24\!\cdots\!00}a^{15}+\frac{40097790517157}{97\!\cdots\!60}a^{14}+\frac{91803341970007}{40\!\cdots\!50}a^{13}-\frac{27\!\cdots\!27}{48\!\cdots\!00}a^{12}+\frac{17\!\cdots\!21}{12\!\cdots\!50}a^{11}-\frac{10487105813189}{19\!\cdots\!50}a^{10}+\frac{50\!\cdots\!53}{24\!\cdots\!00}a^{9}-\frac{41\!\cdots\!53}{80\!\cdots\!00}a^{8}+\frac{44\!\cdots\!47}{24\!\cdots\!00}a^{7}-\frac{77\!\cdots\!99}{40\!\cdots\!50}a^{6}-\frac{418457505322241}{86\!\cdots\!75}a^{5}-\frac{24\!\cdots\!41}{69\!\cdots\!00}a^{4}+\frac{638167300841269}{11\!\cdots\!90}a^{3}-\frac{73\!\cdots\!03}{48\!\cdots\!00}a^{2}+\frac{40\!\cdots\!93}{48\!\cdots\!80}a+\frac{17\!\cdots\!47}{48\!\cdots\!00}$, $\frac{6647322906421}{48\!\cdots\!00}a^{17}-\frac{20084989307}{34\!\cdots\!70}a^{16}-\frac{2813225373191}{23\!\cdots\!00}a^{15}+\frac{954447837589}{97\!\cdots\!16}a^{14}+\frac{384609217880449}{48\!\cdots\!00}a^{13}-\frac{209857531263823}{12\!\cdots\!50}a^{12}+\frac{716576462230427}{24\!\cdots\!00}a^{11}+\frac{73330905780449}{60\!\cdots\!25}a^{10}+\frac{27930745059757}{17\!\cdots\!50}a^{9}-\frac{19\!\cdots\!91}{60\!\cdots\!25}a^{8}+\frac{15\!\cdots\!17}{24\!\cdots\!00}a^{7}-\frac{16\!\cdots\!59}{24\!\cdots\!00}a^{6}-\frac{41\!\cdots\!59}{53\!\cdots\!00}a^{5}+\frac{200175808105858}{86\!\cdots\!75}a^{4}+\frac{76\!\cdots\!07}{97\!\cdots\!60}a^{3}-\frac{18\!\cdots\!99}{24\!\cdots\!00}a^{2}-\frac{12\!\cdots\!17}{32\!\cdots\!20}a+\frac{32\!\cdots\!51}{24\!\cdots\!00}$, $\frac{1218282785753}{24\!\cdots\!90}a^{17}-\frac{6113104177}{924878032614792}a^{16}-\frac{12782630712427}{24\!\cdots\!90}a^{15}+\frac{42331126180871}{97\!\cdots\!60}a^{14}+\frac{164973721966411}{48\!\cdots\!80}a^{13}-\frac{32502242185627}{46\!\cdots\!60}a^{12}+\frac{189886773287719}{24\!\cdots\!90}a^{11}+\frac{85973143755445}{48\!\cdots\!58}a^{10}-\frac{355097761308641}{97\!\cdots\!16}a^{9}-\frac{2199918225589}{13\!\cdots\!88}a^{8}+\frac{36\!\cdots\!17}{48\!\cdots\!80}a^{7}-\frac{10\!\cdots\!47}{24\!\cdots\!90}a^{6}-\frac{13\!\cdots\!17}{24\!\cdots\!90}a^{5}-\frac{10\!\cdots\!07}{97\!\cdots\!60}a^{4}+\frac{29\!\cdots\!49}{48\!\cdots\!80}a^{3}+\frac{51\!\cdots\!65}{64\!\cdots\!44}a^{2}-\frac{11\!\cdots\!41}{34\!\cdots\!70}a+\frac{82\!\cdots\!91}{97\!\cdots\!60}$, $\frac{5606294806037}{69\!\cdots\!00}a^{17}-\frac{57370405459}{346829262230547}a^{16}-\frac{23672498084591}{53\!\cdots\!00}a^{15}+\frac{4202699889347}{24\!\cdots\!29}a^{14}+\frac{716140810951301}{48\!\cdots\!00}a^{13}-\frac{18\!\cdots\!67}{12\!\cdots\!50}a^{12}+\frac{11\!\cdots\!43}{24\!\cdots\!00}a^{11}-\frac{71\!\cdots\!03}{12\!\cdots\!50}a^{10}+\frac{980381265993593}{17\!\cdots\!50}a^{9}-\frac{10\!\cdots\!63}{12\!\cdots\!50}a^{8}+\frac{88\!\cdots\!03}{24\!\cdots\!00}a^{7}-\frac{17\!\cdots\!31}{24\!\cdots\!00}a^{6}+\frac{65\!\cdots\!77}{16\!\cdots\!00}a^{5}-\frac{22\!\cdots\!76}{60\!\cdots\!25}a^{4}+\frac{26\!\cdots\!61}{13\!\cdots\!80}a^{3}-\frac{54\!\cdots\!94}{60\!\cdots\!25}a^{2}+\frac{28\!\cdots\!13}{21\!\cdots\!48}a-\frac{99\!\cdots\!41}{24\!\cdots\!00}$, $\frac{26115068266027}{24\!\cdots\!00}a^{17}-\frac{13485546580909}{97\!\cdots\!60}a^{16}-\frac{1418526710557}{19\!\cdots\!50}a^{15}+\frac{31160631963367}{19\!\cdots\!32}a^{14}+\frac{517523120056769}{12\!\cdots\!50}a^{13}-\frac{83\!\cdots\!33}{48\!\cdots\!00}a^{12}+\frac{50\!\cdots\!99}{12\!\cdots\!50}a^{11}-\frac{45\!\cdots\!73}{12\!\cdots\!50}a^{10}+\frac{17\!\cdots\!27}{24\!\cdots\!00}a^{9}-\frac{50\!\cdots\!61}{24\!\cdots\!00}a^{8}+\frac{15\!\cdots\!83}{24\!\cdots\!00}a^{7}-\frac{14\!\cdots\!69}{17\!\cdots\!50}a^{6}+\frac{27\!\cdots\!51}{80\!\cdots\!00}a^{5}-\frac{38\!\cdots\!23}{48\!\cdots\!00}a^{4}+\frac{11\!\cdots\!49}{48\!\cdots\!80}a^{3}-\frac{11\!\cdots\!77}{48\!\cdots\!00}a^{2}+\frac{21\!\cdots\!59}{17\!\cdots\!40}a-\frac{33\!\cdots\!61}{69\!\cdots\!00}$, $\frac{212787290183}{64\!\cdots\!44}a^{17}+\frac{417112613678}{40\!\cdots\!15}a^{16}-\frac{8561115815173}{32\!\cdots\!20}a^{15}-\frac{529072825028}{13\!\cdots\!05}a^{14}+\frac{88134976075513}{32\!\cdots\!20}a^{13}+\frac{6391040659253}{16\!\cdots\!60}a^{12}-\frac{23608803767767}{53\!\cdots\!20}a^{11}+\frac{315046488569263}{80\!\cdots\!30}a^{10}+\frac{11238061804717}{13\!\cdots\!05}a^{9}+\frac{25947720947728}{809268278537943}a^{8}+\frac{7492411241087}{66606442678020}a^{7}+\frac{61\!\cdots\!03}{16\!\cdots\!60}a^{6}-\frac{95\!\cdots\!97}{32\!\cdots\!20}a^{5}+\frac{1837712873984}{13\!\cdots\!05}a^{4}+\frac{86\!\cdots\!53}{32\!\cdots\!20}a^{3}+\frac{14\!\cdots\!09}{80\!\cdots\!30}a^{2}-\frac{10\!\cdots\!59}{64\!\cdots\!44}a+\frac{79885961784599}{26\!\cdots\!10}$, $\frac{678614344253}{60\!\cdots\!25}a^{17}+\frac{3521486538427}{97\!\cdots\!60}a^{16}-\frac{6676848859529}{80\!\cdots\!00}a^{15}-\frac{22433676833567}{97\!\cdots\!60}a^{14}+\frac{56255849903417}{60\!\cdots\!25}a^{13}+\frac{477243181650857}{48\!\cdots\!00}a^{12}-\frac{288227941443431}{12\!\cdots\!50}a^{11}+\frac{500961832747871}{60\!\cdots\!25}a^{10}+\frac{44\!\cdots\!37}{24\!\cdots\!00}a^{9}-\frac{31\!\cdots\!01}{24\!\cdots\!00}a^{8}+\frac{88\!\cdots\!83}{24\!\cdots\!00}a^{7}+\frac{77\!\cdots\!46}{60\!\cdots\!25}a^{6}-\frac{19\!\cdots\!16}{20\!\cdots\!75}a^{5}-\frac{58\!\cdots\!83}{48\!\cdots\!00}a^{4}+\frac{23\!\cdots\!76}{12\!\cdots\!45}a^{3}-\frac{23\!\cdots\!41}{69\!\cdots\!00}a^{2}-\frac{57\!\cdots\!27}{16\!\cdots\!60}a-\frac{14\!\cdots\!17}{48\!\cdots\!00}$ | 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: | \( 400345.116 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 400345.116 \cdot 9}{2\cdot\sqrt{502592611936843000000000000}}\cr\approx \mathstrut & 1.22647250 \end{aligned}\]
Galois group
A solvable group of order 18 |
The 6 conjugacy class representatives for $C_3^2 : C_2$ |
Character table for $C_3^2 : C_2$ |
Intermediate fields
\(\Q(\sqrt{-43}) \), 3.1.4300.2 x3, 3.1.172.1 x3, 3.1.4300.1 x3, 3.1.1075.1 x3, 6.0.795070000.1, 6.0.1272112.1, 6.0.795070000.2, 6.0.49691875.1, 9.1.3418801000000.2 x9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(5\) | 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(43\) | 43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |