Normalized defining polynomial
\( x^{24} + 6 x^{22} - 4 x^{21} - 99 x^{20} - 21 x^{19} - 269 x^{18} - 162 x^{17} + 2106 x^{16} - 1085 x^{15} + 4752 x^{14} + 12591 x^{13} - 6399 x^{12} + 63306 x^{11} + 91182 x^{10} + 127001 x^{9} + 434943 x^{8} + 326223 x^{7} + 684787 x^{6} + 581175 x^{5} + 631794 x^{4} + 411790 x^{3} + 328707 x^{2} + 134169 x + 46819 \)
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 12]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(10370328622637411153913943764610276201876257=3^{36}\cdot 17^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.99$ | 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: | $3, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(153=3^{2}\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{153}(128,·)$, $\chi_{153}(1,·)$, $\chi_{153}(2,·)$, $\chi_{153}(67,·)$, $\chi_{153}(4,·)$, $\chi_{153}(134,·)$, $\chi_{153}(8,·)$, $\chi_{153}(13,·)$, $\chi_{153}(77,·)$, $\chi_{153}(16,·)$, $\chi_{153}(83,·)$, $\chi_{153}(26,·)$, $\chi_{153}(32,·)$, $\chi_{153}(103,·)$, $\chi_{153}(104,·)$, $\chi_{153}(64,·)$, $\chi_{153}(106,·)$, $\chi_{153}(110,·)$, $\chi_{153}(115,·)$, $\chi_{153}(52,·)$, $\chi_{153}(53,·)$, $\chi_{153}(118,·)$, $\chi_{153}(55,·)$, $\chi_{153}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{47} a^{21} - \frac{20}{47} a^{19} + \frac{18}{47} a^{18} - \frac{2}{47} a^{17} + \frac{11}{47} a^{16} - \frac{9}{47} a^{15} - \frac{22}{47} a^{14} - \frac{3}{47} a^{13} - \frac{6}{47} a^{12} + \frac{22}{47} a^{11} - \frac{9}{47} a^{10} - \frac{6}{47} a^{9} + \frac{10}{47} a^{8} + \frac{7}{47} a^{7} - \frac{9}{47} a^{6} - \frac{4}{47} a^{5} + \frac{8}{47} a^{4} - \frac{3}{47} a^{3} + \frac{6}{47} a^{2} - \frac{8}{47} a - \frac{11}{47}$, $\frac{1}{392165791} a^{22} - \frac{1246293}{392165791} a^{21} - \frac{162383798}{392165791} a^{20} - \frac{150285328}{392165791} a^{19} + \frac{40101587}{392165791} a^{18} - \frac{20964398}{392165791} a^{17} + \frac{102755781}{392165791} a^{16} + \frac{87874179}{392165791} a^{15} + \frac{194196157}{392165791} a^{14} - \frac{159946946}{392165791} a^{13} - \frac{148220060}{392165791} a^{12} - \frac{92149675}{392165791} a^{11} + \frac{15971386}{392165791} a^{10} - \frac{86459158}{392165791} a^{9} - \frac{131429229}{392165791} a^{8} + \frac{141909906}{392165791} a^{7} + \frac{132772686}{392165791} a^{6} - \frac{6713966}{392165791} a^{5} - \frac{139826553}{392165791} a^{4} + \frac{158743287}{392165791} a^{3} - \frac{140891731}{392165791} a^{2} - \frac{116302781}{392165791} a + \frac{27894481}{392165791}$, $\frac{1}{3790690878659584575852095719862284857072457249457420869379299} a^{23} + \frac{3266698025928559952790936276434215454714672118397522}{3790690878659584575852095719862284857072457249457420869379299} a^{22} - \frac{21237435779209829504823114292582909329387755385268318387553}{3790690878659584575852095719862284857072457249457420869379299} a^{21} - \frac{682912350552189334355472621112853718243568374005040876284947}{3790690878659584575852095719862284857072457249457420869379299} a^{20} + \frac{27447527317055326444416539365344902552140327629056928960076}{80652997418289033528767994039623082065371430839519592965517} a^{19} - \frac{362180338240859277182384721183991034812053991425332515321168}{3790690878659584575852095719862284857072457249457420869379299} a^{18} + \frac{395775931720454730013119435410339720457533238063589189093872}{3790690878659584575852095719862284857072457249457420869379299} a^{17} - \frac{1887745710745735241164091867758632122285165014441713141368630}{3790690878659584575852095719862284857072457249457420869379299} a^{16} - \frac{233962317924050374550121479166514219731636492345770721018241}{3790690878659584575852095719862284857072457249457420869379299} a^{15} + \frac{25049423885935693933919986106697974049156041325182729500851}{80652997418289033528767994039623082065371430839519592965517} a^{14} + \frac{1468638854041361955790561328968961748003755636417602278323500}{3790690878659584575852095719862284857072457249457420869379299} a^{13} + \frac{1367575022560757878862251575126156625298257955974096067863975}{3790690878659584575852095719862284857072457249457420869379299} a^{12} + \frac{318898367672296046941502951924006534582208210744142009854241}{3790690878659584575852095719862284857072457249457420869379299} a^{11} + \frac{1487906365463741914425864066570565892026266295496018321346517}{3790690878659584575852095719862284857072457249457420869379299} a^{10} + \frac{788429854216993764595256738467296518987649642358029575865561}{3790690878659584575852095719862284857072457249457420869379299} a^{9} - \frac{1193689759688778478350764598350250586432575428212109400958695}{3790690878659584575852095719862284857072457249457420869379299} a^{8} - \frac{1841068608221735249342232569954017022878246282026516985604829}{3790690878659584575852095719862284857072457249457420869379299} a^{7} - \frac{386850524535931943317326941523740819616398724294901561021941}{3790690878659584575852095719862284857072457249457420869379299} a^{6} + \frac{1270989428145675486493525718410838310073264763405127083703416}{3790690878659584575852095719862284857072457249457420869379299} a^{5} + \frac{1839237713861598744778628549063117180869845627712533270583780}{3790690878659584575852095719862284857072457249457420869379299} a^{4} - \frac{1538127303883097117001588285604328851090101689850532771183520}{3790690878659584575852095719862284857072457249457420869379299} a^{3} - \frac{1522195808247345943979909447761992780642034425115004603806569}{3790690878659584575852095719862284857072457249457420869379299} a^{2} - \frac{256234361319933948335518039193145097468113873770522959253282}{3790690878659584575852095719862284857072457249457420869379299} a + \frac{271313780248774532340668121459730344777286892188197225405132}{3790690878659584575852095719862284857072457249457420869379299}$
Class group and class number
$C_{2}\times C_{22}\times C_{22}$, which has order $968$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 363126034.10126877 \) (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
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 4.4.4913.1, 6.6.32234193.1, 8.0.33237432513.1, 12.12.5104819233548816337.1 |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | R | $24$ | $24$ | $24$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ | $24$ | $24$ | $24$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ | $24$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $17$ | 17.8.7.1 | $x^{8} - 1377$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.1 | $x^{8} - 1377$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
| 17.8.7.1 | $x^{8} - 1377$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |