Normalized defining polynomial
\( x^{18} - 8 x^{17} + 33 x^{16} - 114 x^{15} + 383 x^{14} - 1156 x^{13} + 3191 x^{12} - 8326 x^{11} + \cdots + 2736 \)
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: | \(-18294428062468623673667584\) \(\medspace = -\,2^{12}\cdot 7^{12}\cdot 19^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.32\) | 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: | $2^{2/3}7^{2/3}19^{1/2}\approx 25.319909997284967$ | ||
Ramified primes: | \(2\), \(7\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{12}a^{9}-\frac{1}{12}a^{8}-\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{5}{12}a^{5}+\frac{5}{12}a^{4}+\frac{1}{3}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{8}-\frac{1}{6}a^{7}-\frac{1}{12}a^{6}-\frac{1}{6}a^{5}+\frac{5}{12}a^{4}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{8}+\frac{1}{12}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{12}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}a$, $\frac{1}{24}a^{12}-\frac{1}{24}a^{10}-\frac{1}{24}a^{8}+\frac{1}{6}a^{7}+\frac{5}{24}a^{6}-\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{24}a^{13}-\frac{1}{24}a^{11}-\frac{1}{24}a^{9}-\frac{1}{8}a^{7}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{144}a^{14}+\frac{1}{72}a^{13}-\frac{1}{48}a^{12}+\frac{1}{36}a^{11}+\frac{1}{48}a^{10}-\frac{1}{72}a^{9}+\frac{11}{144}a^{8}+\frac{1}{36}a^{7}+\frac{7}{36}a^{6}-\frac{7}{36}a^{5}-\frac{13}{36}a^{4}-\frac{1}{36}a^{2}$, $\frac{1}{1440}a^{15}+\frac{1}{360}a^{14}+\frac{5}{288}a^{13}+\frac{1}{90}a^{12}-\frac{49}{1440}a^{11}-\frac{5}{144}a^{10}-\frac{41}{1440}a^{9}-\frac{2}{45}a^{8}-\frac{103}{720}a^{6}-\frac{23}{120}a^{5}+\frac{7}{90}a^{4}+\frac{113}{360}a^{3}+\frac{31}{90}a^{2}+\frac{1}{3}a+\frac{3}{20}$, $\frac{1}{4320}a^{16}-\frac{11}{4320}a^{14}-\frac{1}{1080}a^{13}+\frac{7}{4320}a^{12}+\frac{11}{720}a^{11}-\frac{3}{160}a^{10}+\frac{1}{216}a^{9}-\frac{11}{180}a^{8}+\frac{457}{2160}a^{7}-\frac{43}{180}a^{6}+\frac{11}{45}a^{5}-\frac{17}{40}a^{4}+\frac{83}{270}a^{3}+\frac{53}{135}a^{2}+\frac{29}{180}a-\frac{1}{30}$, $\frac{1}{5841255867840}a^{17}-\frac{61264333}{584125586784}a^{16}+\frac{666851251}{5841255867840}a^{15}-\frac{3689448223}{2920627933920}a^{14}-\frac{116236643443}{5841255867840}a^{13}+\frac{2648915627}{1460313966960}a^{12}+\frac{54238309007}{1947085289280}a^{11}+\frac{5409473597}{584125586784}a^{10}+\frac{60853489577}{2920627933920}a^{9}-\frac{211857668867}{2920627933920}a^{8}-\frac{14143562783}{1460313966960}a^{7}+\frac{90660966097}{486771322320}a^{6}-\frac{53075472473}{162257107440}a^{5}-\frac{30236583101}{730156983480}a^{4}+\frac{71699295749}{146031396696}a^{3}+\frac{268725277531}{730156983480}a^{2}+\frac{596899219}{3202442910}a-\frac{490221533}{2134961940}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{11093517673}{1947085289280}a^{17}-\frac{31837538681}{973542644640}a^{16}+\frac{183307987619}{1947085289280}a^{15}-\frac{291733809431}{973542644640}a^{14}+\frac{659509070143}{649028429760}a^{13}-\frac{144221894429}{54085702480}a^{12}+\frac{13040231010797}{1947085289280}a^{11}-\frac{1792620978951}{108171404960}a^{10}+\frac{30351912636253}{973542644640}a^{9}-\frac{45673482685579}{973542644640}a^{8}+\frac{38529028885909}{486771322320}a^{7}-\frac{11743798210981}{97354264464}a^{6}+\frac{11772887448029}{162257107440}a^{5}+\frac{1967369994013}{16225710744}a^{4}-\frac{73729101211723}{243385661160}a^{3}+\frac{23286365724577}{81128553720}a^{2}-\frac{144679520551}{1067480970}a+\frac{3844336747}{142330796}$, $\frac{11093517673}{1947085289280}a^{17}-\frac{31837538681}{973542644640}a^{16}+\frac{183307987619}{1947085289280}a^{15}-\frac{291733809431}{973542644640}a^{14}+\frac{659509070143}{649028429760}a^{13}-\frac{144221894429}{54085702480}a^{12}+\frac{13040231010797}{1947085289280}a^{11}-\frac{1792620978951}{108171404960}a^{10}+\frac{30351912636253}{973542644640}a^{9}-\frac{45673482685579}{973542644640}a^{8}+\frac{38529028885909}{486771322320}a^{7}-\frac{11743798210981}{97354264464}a^{6}+\frac{11772887448029}{162257107440}a^{5}+\frac{1967369994013}{16225710744}a^{4}-\frac{73729101211723}{243385661160}a^{3}+\frac{23286365724577}{81128553720}a^{2}-\frac{144679520551}{1067480970}a+\frac{3986667543}{142330796}$, $\frac{88014253}{4921024320}a^{17}-\frac{143858669}{1230256080}a^{16}+\frac{2041870939}{4921024320}a^{15}-\frac{578445509}{410085360}a^{14}+\frac{1549916611}{328068288}a^{13}-\frac{33328729411}{2460512160}a^{12}+\frac{59927738651}{1640341440}a^{11}-\frac{57561619343}{615128040}a^{10}+\frac{505026927643}{2460512160}a^{9}-\frac{966743645921}{2460512160}a^{8}+\frac{22221808811}{30756402}a^{7}-\frac{506045350691}{410085360}a^{6}+\frac{707736533317}{410085360}a^{5}-\frac{557314701691}{307564020}a^{4}+\frac{92965791587}{68347560}a^{3}-\frac{430649941447}{615128040}a^{2}+\frac{1217956441}{5395860}a-\frac{65518399}{1798620}$, $\frac{13001}{4177440}a^{17}-\frac{94909}{1392480}a^{16}+\frac{1672013}{4177440}a^{15}-\frac{5925641}{4177440}a^{14}+\frac{3945247}{835488}a^{13}-\frac{21533563}{1392480}a^{12}+\frac{61505047}{1392480}a^{11}-\frac{489851843}{4177440}a^{10}+\frac{102431291}{348120}a^{9}-\frac{664851641}{1044360}a^{8}+\frac{166724465}{139248}a^{7}-\frac{375848291}{174060}a^{6}+\frac{1264468399}{348120}a^{5}-\frac{5132546513}{1044360}a^{4}+\frac{1245319163}{261090}a^{3}-\frac{89259493}{29010}a^{2}+\frac{69123527}{58020}a-\frac{2079469}{9670}$, $\frac{27438292973}{730156983480}a^{17}-\frac{124900799413}{486771322320}a^{16}+\frac{1340988936043}{1460313966960}a^{15}-\frac{1118631497911}{365078491740}a^{14}+\frac{15030453322513}{1460313966960}a^{13}-\frac{3613404389249}{121692830580}a^{12}+\frac{38668374014893}{486771322320}a^{11}-\frac{74279142170599}{365078491740}a^{10}+\frac{72623176664753}{162257107440}a^{9}-\frac{12\!\cdots\!81}{1460313966960}a^{8}+\frac{375009000346847}{243385661160}a^{7}-\frac{32143726933559}{12169283058}a^{6}+\frac{445623839164393}{121692830580}a^{5}-\frac{66877247361923}{18253924587}a^{4}+\frac{904217354685913}{365078491740}a^{3}-\frac{128282315404297}{121692830580}a^{2}+\frac{264036897109}{1067480970}a-\frac{716960303}{35582699}$, $\frac{126861892847}{5841255867840}a^{17}-\frac{347320647167}{2920627933920}a^{16}+\frac{1945272008699}{5841255867840}a^{15}-\frac{209048756003}{194708528928}a^{14}+\frac{1411897864367}{389417057856}a^{13}-\frac{2730088805807}{292062793392}a^{12}+\frac{1017165281203}{43268561984}a^{11}-\frac{168787901360023}{2920627933920}a^{10}+\frac{19292835734663}{182539245870}a^{9}-\frac{90736622480609}{584125586784}a^{8}+\frac{76719960986963}{292062793392}a^{7}-\frac{15627479509669}{40564276860}a^{6}+\frac{81746326810469}{486771322320}a^{5}+\frac{398065681952999}{730156983480}a^{4}-\frac{141847893410981}{121692830580}a^{3}+\frac{781395420644533}{730156983480}a^{2}-\frac{814575975694}{1601221455}a+\frac{58444915408}{533740485}$, $\frac{1247535713}{162257107440}a^{17}-\frac{25198438601}{486771322320}a^{16}+\frac{17956116233}{97354264464}a^{15}-\frac{299802256939}{486771322320}a^{14}+\frac{335130914579}{162257107440}a^{13}-\frac{2893233700279}{486771322320}a^{12}+\frac{7739435895419}{486771322320}a^{11}-\frac{19787368666229}{486771322320}a^{10}+\frac{120520361431}{1352142562}a^{9}-\frac{10238812154537}{60846415290}a^{8}+\frac{18594368048191}{60846415290}a^{7}-\frac{63533710128617}{121692830580}a^{6}+\frac{7298635049942}{10141069215}a^{5}-\frac{43259861418883}{60846415290}a^{4}+\frac{14127333782689}{30423207645}a^{3}-\frac{10917653243171}{60846415290}a^{2}+\frac{10809760451}{355826990}a+\frac{199583758}{177913495}$, $\frac{5848777097}{1168251173568}a^{17}-\frac{4865887687}{584125586784}a^{16}-\frac{47435261269}{1168251173568}a^{15}+\frac{85314894985}{584125586784}a^{14}-\frac{577125759515}{1168251173568}a^{13}+\frac{670320884497}{292062793392}a^{12}-\frac{2748579755833}{389417057856}a^{11}+\frac{11669775641681}{584125586784}a^{10}-\frac{35172976952207}{584125586784}a^{9}+\frac{84215903977679}{584125586784}a^{8}-\frac{79755797897395}{292062793392}a^{7}+\frac{16852391064635}{32451421488}a^{6}-\frac{95546627767651}{97354264464}a^{5}+\frac{213404326350821}{146031396696}a^{4}-\frac{218983431218551}{146031396696}a^{3}+\frac{144086176094465}{146031396696}a^{2}-\frac{121274337974}{320244291}a+\frac{27673339367}{426992388}$ | 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: | \( 229812.209675 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 229812.209675 \cdot 3}{2\cdot\sqrt{18294428062468623673667584}}\cr\approx \mathstrut & 1.23005280159 \end{aligned}\]
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
\(\Q(\sqrt{-19}) \), 3.1.3724.2 x3, 3.1.76.1 x3, 3.1.931.1 x3, 3.1.3724.1 x3, 6.0.263495344.2, 6.0.109744.2, 6.0.16468459.2, 6.0.263495344.1, 9.1.981256661056.1 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 | ${\href{/padicField/3.2.0.1}{2} }^{9}$ | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |