Normalized defining polynomial
\( x^{19} - 9 x^{18} + 24 x^{17} - 6 x^{16} - 42 x^{15} + 4 x^{14} + 34 x^{13} + 74 x^{12} + 46 x^{11} - 238 x^{10} - 49 x^{9} + 181 x^{8} + 195 x^{7} + 116 x^{6} - 193 x^{5} - 275 x^{4} + 106 x^{3} + \cdots - 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-49578494467761916312526550343\) \(\medspace = -\,1543^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.38\) | 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: | $1543^{1/2}\approx 39.28103868280471$ | ||
Ramified primes: | \(1543\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1543}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{12}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{2}{9}a^{6}+\frac{2}{9}a^{5}-\frac{2}{9}a^{4}-\frac{2}{9}a^{2}-\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{2}{9}a^{7}+\frac{2}{9}a^{5}+\frac{2}{9}a^{4}-\frac{2}{9}a^{3}-\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{855}a^{16}-\frac{11}{285}a^{15}+\frac{41}{855}a^{14}+\frac{89}{855}a^{13}+\frac{14}{171}a^{12}+\frac{11}{285}a^{11}+\frac{103}{855}a^{10}+\frac{82}{855}a^{9}-\frac{4}{57}a^{8}-\frac{67}{285}a^{7}-\frac{17}{855}a^{6}-\frac{371}{855}a^{5}+\frac{73}{171}a^{4}-\frac{29}{285}a^{3}+\frac{251}{855}a^{2}-\frac{169}{855}a-\frac{61}{855}$, $\frac{1}{2565}a^{17}-\frac{98}{2565}a^{15}-\frac{26}{855}a^{14}-\frac{413}{2565}a^{13}-\frac{32}{2565}a^{12}-\frac{43}{2565}a^{11}-\frac{43}{855}a^{10}+\frac{122}{855}a^{9}+\frac{194}{2565}a^{8}-\frac{5}{27}a^{7}-\frac{184}{855}a^{6}+\frac{92}{2565}a^{5}-\frac{487}{2565}a^{4}+\frac{179}{513}a^{3}+\frac{298}{855}a^{2}+\frac{917}{2565}a+\frac{1027}{2565}$, $\frac{1}{4772442701295}a^{18}+\frac{662887919}{4772442701295}a^{17}+\frac{1752360862}{4772442701295}a^{16}+\frac{33487901806}{954488540259}a^{15}+\frac{13895980616}{954488540259}a^{14}+\frac{58546782047}{1590814233765}a^{13}+\frac{736040120899}{4772442701295}a^{12}-\frac{723266576861}{4772442701295}a^{11}-\frac{43584455216}{318162846753}a^{10}+\frac{474576464078}{4772442701295}a^{9}+\frac{935269937}{38800347165}a^{8}+\frac{647391070363}{4772442701295}a^{7}-\frac{1487278745026}{4772442701295}a^{6}-\frac{660860588378}{1590814233765}a^{5}-\frac{1990918487548}{4772442701295}a^{4}+\frac{326430003014}{4772442701295}a^{3}-\frac{26896025713}{60410667105}a^{2}+\frac{185213667625}{954488540259}a-\frac{51519525688}{251181194805}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $\frac{17959875136}{530271411255}a^{18}-\frac{170742520601}{530271411255}a^{17}+\frac{168526829131}{176757137085}a^{16}-\frac{249098257613}{530271411255}a^{15}-\frac{107705448476}{58919045695}a^{14}+\frac{141861836681}{106054282251}a^{13}+\frac{1026407780539}{530271411255}a^{12}+\frac{196483523537}{530271411255}a^{11}-\frac{213101128832}{530271411255}a^{10}-\frac{256272399431}{35351427417}a^{9}+\frac{32404310506}{12933449055}a^{8}+\frac{4995260337812}{530271411255}a^{7}-\frac{8235336552}{58919045695}a^{6}-\frac{419460757742}{106054282251}a^{5}-\frac{1360146353143}{530271411255}a^{4}-\frac{1658717815748}{530271411255}a^{3}+\frac{1831912804}{353278755}a^{2}-\frac{623739678769}{530271411255}a+\frac{507780115297}{530271411255}$, $\frac{13156927406}{4772442701295}a^{18}-\frac{29591640715}{954488540259}a^{17}+\frac{109483527277}{954488540259}a^{16}-\frac{461920476377}{4772442701295}a^{15}-\frac{258869780084}{954488540259}a^{14}+\frac{4827003128}{16745412987}a^{13}+\frac{3067488956732}{4772442701295}a^{12}-\frac{277717320521}{954488540259}a^{11}-\frac{258386622394}{318162846753}a^{10}-\frac{1551479184340}{954488540259}a^{9}+\frac{11331086483}{7760069433}a^{8}+\frac{3106736399332}{954488540259}a^{7}+\frac{1419688731886}{4772442701295}a^{6}-\frac{1024009356113}{318162846753}a^{5}-\frac{2603687341768}{954488540259}a^{4}-\frac{3058554546107}{4772442701295}a^{3}+\frac{2342384036}{635901759}a^{2}+\frac{991610363252}{954488540259}a+\frac{1824113337292}{4772442701295}$, $\frac{42669179771}{4772442701295}a^{18}-\frac{334653386162}{4772442701295}a^{17}+\frac{596805388544}{4772442701295}a^{16}+\frac{848855610022}{4772442701295}a^{15}-\frac{25025270768}{50236238961}a^{14}+\frac{3735872006}{83727064935}a^{13}-\frac{1561202891824}{4772442701295}a^{12}+\frac{295109856557}{251181194805}a^{11}+\frac{330932984561}{318162846753}a^{10}-\frac{5562234879449}{4772442701295}a^{9}-\frac{84517697401}{38800347165}a^{8}+\frac{5383654096436}{4772442701295}a^{7}+\frac{7442914958137}{4772442701295}a^{6}+\frac{7209378381409}{1590814233765}a^{5}-\frac{4074757663706}{4772442701295}a^{4}-\frac{1972104993328}{954488540259}a^{3}+\frac{286539964}{60410667105}a^{2}+\frac{472190295728}{954488540259}a+\frac{3527831449339}{4772442701295}$, $\frac{179374741}{110987039565}a^{18}-\frac{3701085019}{110987039565}a^{17}+\frac{25590312451}{110987039565}a^{16}-\frac{71663645878}{110987039565}a^{15}+\frac{44860567228}{110987039565}a^{14}+\frac{47022721987}{36995679855}a^{13}-\frac{35277665647}{22197407913}a^{12}-\frac{24021131401}{22197407913}a^{11}+\frac{41117508368}{36995679855}a^{10}+\frac{781793777}{5841423135}a^{9}+\frac{3609172273}{902333655}a^{8}-\frac{292017712856}{110987039565}a^{7}-\frac{36868288612}{5841423135}a^{6}+\frac{162142107617}{36995679855}a^{5}+\frac{409488479663}{110987039565}a^{4}-\frac{99131109319}{110987039565}a^{3}+\frac{809404279}{280979847}a^{2}-\frac{180538537792}{110987039565}a+\frac{108776696858}{110987039565}$, $\frac{1213365566}{78236765595}a^{18}-\frac{11519989592}{78236765595}a^{17}+\frac{34715751704}{78236765595}a^{16}-\frac{23264664908}{78236765595}a^{15}-\frac{9002950439}{15647353119}a^{14}+\frac{13450168954}{26078921865}a^{13}+\frac{5510932091}{78236765595}a^{12}+\frac{95301531908}{78236765595}a^{11}-\frac{344972290}{5215784373}a^{10}-\frac{252866301329}{78236765595}a^{9}+\frac{394527154}{636071265}a^{8}+\frac{221715247346}{78236765595}a^{7}+\frac{35067046522}{78236765595}a^{6}+\frac{50389294454}{26078921865}a^{5}-\frac{316847425331}{78236765595}a^{4}-\frac{13898712415}{15647353119}a^{3}+\frac{1318508029}{990338805}a^{2}-\frac{6794209435}{15647353119}a+\frac{75077899054}{78236765595}$, $\frac{21587547676}{4772442701295}a^{18}-\frac{204093079054}{4772442701295}a^{17}+\frac{125740279940}{954488540259}a^{16}-\frac{123915700130}{954488540259}a^{15}+\frac{133182683302}{4772442701295}a^{14}-\frac{58359328667}{530271411255}a^{13}-\frac{15377793235}{50236238961}a^{12}+\frac{6248593061482}{4772442701295}a^{11}-\frac{321221377918}{1590814233765}a^{10}-\frac{6071394919324}{4772442701295}a^{9}-\frac{1926935149}{12933449055}a^{8}-\frac{792817402987}{954488540259}a^{7}+\frac{14579637031844}{4772442701295}a^{6}+\frac{241896317311}{176757137085}a^{5}-\frac{14963876953072}{4772442701295}a^{4}-\frac{6374647569502}{4772442701295}a^{3}-\frac{52486314469}{60410667105}a^{2}+\frac{1224540768557}{4772442701295}a+\frac{3227493096854}{4772442701295}$, $\frac{47936257897}{4772442701295}a^{18}-\frac{448044733822}{4772442701295}a^{17}+\frac{1202302108546}{4772442701295}a^{16}+\frac{294510683534}{4772442701295}a^{15}-\frac{5068119670088}{4772442701295}a^{14}+\frac{186958326428}{318162846753}a^{13}+\frac{9100301884528}{4772442701295}a^{12}-\frac{6106874069861}{4772442701295}a^{11}-\frac{2721805230883}{1590814233765}a^{10}-\frac{82914318008}{954488540259}a^{9}+\frac{53670820609}{38800347165}a^{8}+\frac{819778705741}{251181194805}a^{7}-\frac{292468243294}{251181194805}a^{6}-\frac{1305481878095}{318162846753}a^{5}+\frac{3873241031444}{4772442701295}a^{4}+\frac{6368708899649}{4772442701295}a^{3}+\frac{60637748177}{60410667105}a^{2}+\frac{3555906193187}{4772442701295}a-\frac{3267898875646}{4772442701295}$, $\frac{36486838}{60410667105}a^{18}-\frac{904109332}{60410667105}a^{17}+\frac{5679276124}{60410667105}a^{16}-\frac{10611709492}{60410667105}a^{15}-\frac{1560990148}{12082133421}a^{14}+\frac{3401553518}{6712296345}a^{13}+\frac{1864371938}{12082133421}a^{12}-\frac{32509293452}{60410667105}a^{11}-\frac{1785516857}{4027377807}a^{10}-\frac{41513809339}{60410667105}a^{9}+\frac{265006868}{163714545}a^{8}+\frac{79548617746}{60410667105}a^{7}-\frac{125138019436}{60410667105}a^{6}-\frac{13972983007}{6712296345}a^{5}-\frac{26084257816}{60410667105}a^{4}+\frac{7178994551}{60410667105}a^{3}+\frac{136723395311}{60410667105}a^{2}-\frac{6787950947}{12082133421}a-\frac{6261296407}{60410667105}$, $a$ (assuming GRH) | 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: | \( 17317158.8896 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 17317158.8896 \cdot 1}{2\cdot\sqrt{49578494467761916312526550343}}\cr\approx \mathstrut & 1.18699472696 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 38 |
The 11 conjugacy class representatives for $D_{19}$ |
Character table for $D_{19}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $19$ | ${\href{/padicField/3.2.0.1}{2} }^{9}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.2.0.1}{2} }^{9}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19$ | ${\href{/padicField/11.2.0.1}{2} }^{9}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $19$ | $19$ | $19$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1543\) | $\Q_{1543}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |