Normalized defining polynomial
\( x^{24} - 5 x^{23} - 109 x^{22} + 414 x^{21} + 5469 x^{20} - 13112 x^{19} - 160371 x^{18} + 170477 x^{17} + 2897736 x^{16} + 189772 x^{15} - 31755107 x^{14} - 31766523 x^{13} + 194940086 x^{12} + 370716475 x^{11} - 513253877 x^{10} - 1808188459 x^{9} - 371523524 x^{8} + 3300112019 x^{7} + 3631771947 x^{6} + 101680474 x^{5} - 1772648953 x^{4} - 835277336 x^{3} + 26771975 x^{2} + 81314415 x + 12047969 \)
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[24, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(852422557828050325453629703955159792180445490817297=17^{21}\cdot 37^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $132.47$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $17, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(629=17\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{629}(1,·)$, $\chi_{629}(322,·)$, $\chi_{629}(195,·)$, $\chi_{629}(519,·)$, $\chi_{629}(137,·)$, $\chi_{629}(528,·)$, $\chi_{629}(593,·)$, $\chi_{629}(84,·)$, $\chi_{629}(149,·)$, $\chi_{629}(26,·)$, $\chi_{629}(285,·)$, $\chi_{629}(223,·)$, $\chi_{629}(417,·)$, $\chi_{629}(100,·)$, $\chi_{629}(38,·)$, $\chi_{629}(359,·)$, $\chi_{629}(297,·)$, $\chi_{629}(491,·)$, $\chi_{629}(174,·)$, $\chi_{629}(47,·)$, $\chi_{629}(433,·)$, $\chi_{629}(565,·)$, $\chi_{629}(121,·)$, $\chi_{629}(186,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{101} a^{21} - \frac{47}{101} a^{20} + \frac{10}{101} a^{19} - \frac{8}{101} a^{18} - \frac{2}{101} a^{17} - \frac{44}{101} a^{16} + \frac{10}{101} a^{15} + \frac{33}{101} a^{14} - \frac{48}{101} a^{13} + \frac{43}{101} a^{12} - \frac{14}{101} a^{11} + \frac{45}{101} a^{10} + \frac{6}{101} a^{9} + \frac{50}{101} a^{8} - \frac{24}{101} a^{7} + \frac{21}{101} a^{6} + \frac{45}{101} a^{5} - \frac{28}{101} a^{4} + \frac{26}{101} a^{3} + \frac{12}{101} a^{2} + \frac{41}{101} a - \frac{26}{101}$, $\frac{1}{1425211} a^{22} - \frac{160}{1425211} a^{21} + \frac{33904}{1425211} a^{20} - \frac{505835}{1425211} a^{19} - \frac{50810}{1425211} a^{18} + \frac{438724}{1425211} a^{17} - \frac{297008}{1425211} a^{16} - \frac{378433}{1425211} a^{15} + \frac{171862}{1425211} a^{14} - \frac{286726}{1425211} a^{13} - \frac{57292}{1425211} a^{12} - \frac{180880}{1425211} a^{11} - \frac{212331}{1425211} a^{10} - \frac{266157}{1425211} a^{9} - \frac{379273}{1425211} a^{8} + \frac{298663}{1425211} a^{7} + \frac{597309}{1425211} a^{6} + \frac{600483}{1425211} a^{5} - \frac{271126}{1425211} a^{4} + \frac{30909}{1425211} a^{3} - \frac{57572}{1425211} a^{2} - \frac{686207}{1425211} a - \frac{503678}{1425211}$, $\frac{1}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{23} + \frac{22909592498515533802471732752874472960751905753631958146798145223439819039584}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{22} - \frac{242014258189716155864285818857102311795066738703534970583047030047009221545320308}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{21} + \frac{27015910968069735343766302161257615660411984303518084006286745256562885932728826827}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{20} - \frac{8351949634584142643951795583189560761061459142824481755514073794765580344267005860}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{19} - \frac{3216791406223088718521518124131027764447976623640238140907188602427470511352561403}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{18} - \frac{26529849657669959485448026027616745390134152271215580261792478354264777284064878539}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{17} - \frac{19145020642854480780516534703860055004283730883233526287348857311603133566416828982}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{16} - \frac{2115088837811228139990166565560763577236491905172968582536970762508468343688671786}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{15} - \frac{8671594160727062861943366014425883472610558951007005745772296567923284452073013908}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{14} - \frac{14473676983084407897977034193323771407160519619164205799167237094753746616447848917}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{13} + \frac{14907308131107098291057076273434113585046212401223553377900817330490823832996354001}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{12} - \frac{7865961517904501914365097311443306235736341365724285705843727081212104402946408331}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{11} + \frac{15668424806802867066753560784221142196766170018122802634511577312055153173621652612}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{10} + \frac{19888265523873219197125518856008140988760002057365971953236895818483741639377331898}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{9} + \frac{7422644297065387500474263004808603849934577329073580707448384682675137391390467641}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{8} + \frac{23357284702774069143383669962538693926444712838231976687260820767424449574478958187}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{7} - \frac{26267444530232339360410440336706582216481439018831661983522383564116005012530959759}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{6} + \frac{29002058968881411733215547326166041208239517258811281632862160476333039014946622848}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{5} - \frac{11195141368887956140540229408127019685124483264500039190928120725359216342574936267}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{4} - \frac{29797048162092023117152433873080924434596755658250847686935280341597026042675885287}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{3} - \frac{11106206803844337491761258022306965953408806181429032249574797160639029367878459439}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a^{2} - \frac{27949607891834231798043804694836162310256191418134968304266553704691456375682061000}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923} a - \frac{6125441636957724450795583984418114134895031769514896192160137515997118407543062223}{65994022642130944567068461305385276071577248789876743455107588042355715757023885923}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $23$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 655137207724264700 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 24 |
| The 24 conjugacy class representatives for $C_{24}$ |
| Character table for $C_{24}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), 3.3.1369.1, 4.4.4913.1, 6.6.9207752993.1, \(\Q(\zeta_{17})^+\), 12.12.416537479679833550394737.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | $24$ | $24$ | $24$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ | R | $24$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 17 | Data not computed | ||||||
| 37 | Data not computed | ||||||