Normalized defining polynomial
\( x^{26} - x^{25} + 25 x^{24} - 14 x^{23} + 405 x^{22} - 192 x^{21} + 3603 x^{20} - 1111 x^{19} + 22650 x^{18} - 5414 x^{17} + 87624 x^{16} - 5096 x^{15} + 234451 x^{14} - 19756 x^{13} + 367484 x^{12} - 30562 x^{11} + 404986 x^{10} - 57146 x^{9} + 232119 x^{8} - 34742 x^{7} + 91429 x^{6} - 16226 x^{5} + 7319 x^{4} + 98 x^{3} + 168 x^{2} - 10 x + 1 \)
Invariants
| Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-384766437057818380952237905666104641217782272563\)\(\medspace = -\,3^{13}\cdot 53^{24}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $67.64$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $3, 53$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $26$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(159=3\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{159}(1,·)$, $\chi_{159}(130,·)$, $\chi_{159}(68,·)$, $\chi_{159}(134,·)$, $\chi_{159}(10,·)$, $\chi_{159}(13,·)$, $\chi_{159}(142,·)$, $\chi_{159}(77,·)$, $\chi_{159}(16,·)$, $\chi_{159}(148,·)$, $\chi_{159}(152,·)$, $\chi_{159}(89,·)$, $\chi_{159}(155,·)$, $\chi_{159}(28,·)$, $\chi_{159}(95,·)$, $\chi_{159}(97,·)$, $\chi_{159}(100,·)$, $\chi_{159}(107,·)$, $\chi_{159}(44,·)$, $\chi_{159}(46,·)$, $\chi_{159}(47,·)$, $\chi_{159}(49,·)$, $\chi_{159}(116,·)$, $\chi_{159}(119,·)$, $\chi_{159}(121,·)$, $\chi_{159}(122,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22} - \frac{3}{23} a^{21} - \frac{8}{23} a^{20} + \frac{5}{23} a^{19} + \frac{8}{23} a^{18} - \frac{2}{23} a^{17} - \frac{10}{23} a^{16} - \frac{10}{23} a^{14} - \frac{10}{23} a^{13} + \frac{4}{23} a^{12} - \frac{9}{23} a^{11} - \frac{7}{23} a^{10} - \frac{7}{23} a^{9} - \frac{9}{23} a^{8} - \frac{2}{23} a^{7} + \frac{1}{23} a^{6} + \frac{3}{23} a^{5} + \frac{10}{23} a^{4} + \frac{2}{23} a^{3} - \frac{2}{23} a^{2} - \frac{1}{23} a + \frac{4}{23}$, $\frac{1}{23} a^{23} + \frac{6}{23} a^{21} + \frac{4}{23} a^{20} - \frac{1}{23} a^{18} + \frac{7}{23} a^{17} - \frac{7}{23} a^{16} - \frac{10}{23} a^{15} + \frac{6}{23} a^{14} - \frac{3}{23} a^{13} + \frac{3}{23} a^{12} - \frac{11}{23} a^{11} - \frac{5}{23} a^{10} - \frac{7}{23} a^{9} - \frac{6}{23} a^{8} - \frac{5}{23} a^{7} + \frac{6}{23} a^{6} - \frac{4}{23} a^{5} + \frac{9}{23} a^{4} + \frac{4}{23} a^{3} - \frac{7}{23} a^{2} + \frac{1}{23} a - \frac{11}{23}$, $\frac{1}{435105007} a^{24} - \frac{6511450}{435105007} a^{23} + \frac{6511474}{435105007} a^{22} - \frac{21353795}{435105007} a^{21} - \frac{29492337}{435105007} a^{20} + \frac{7684633}{435105007} a^{19} + \frac{170977751}{435105007} a^{18} + \frac{23875413}{435105007} a^{17} - \frac{175808914}{435105007} a^{16} + \frac{163443991}{435105007} a^{15} - \frac{35916163}{435105007} a^{14} - \frac{533332}{435105007} a^{13} - \frac{15687192}{435105007} a^{12} + \frac{138182201}{435105007} a^{11} - \frac{189374544}{435105007} a^{10} - \frac{197759169}{435105007} a^{9} - \frac{53900542}{435105007} a^{8} - \frac{11497280}{435105007} a^{7} - \frac{7263133}{18917609} a^{6} + \frac{179207798}{435105007} a^{5} + \frac{25522905}{435105007} a^{4} + \frac{11250547}{435105007} a^{3} + \frac{131851964}{435105007} a^{2} + \frac{183386491}{435105007} a + \frac{16110395}{435105007}$, $\frac{1}{1365220377829828203115595168074635041613268941514172023597} a^{25} - \frac{406017783584370443327565549872701562558026047166}{1365220377829828203115595168074635041613268941514172023597} a^{24} - \frac{6904047626912328110876965887822271084240905796235622477}{1365220377829828203115595168074635041613268941514172023597} a^{23} + \frac{29323593610972396322021368460009649846832269138918548607}{1365220377829828203115595168074635041613268941514172023597} a^{22} - \frac{135515756698165995059639584286112323307037142623889494750}{1365220377829828203115595168074635041613268941514172023597} a^{21} - \frac{615107161698357627770812767700755334699834287636261025630}{1365220377829828203115595168074635041613268941514172023597} a^{20} - \frac{672829308607094425937406720507672630890146566540313355588}{1365220377829828203115595168074635041613268941514172023597} a^{19} - \frac{625358850504638576360859254186594649583369533651686212516}{1365220377829828203115595168074635041613268941514172023597} a^{18} - \frac{375407253104663712206007379032769987991042808218546838557}{1365220377829828203115595168074635041613268941514172023597} a^{17} + \frac{43394179261886025017011280587476349277189074768710654121}{1365220377829828203115595168074635041613268941514172023597} a^{16} - \frac{67072177062589755015214068849110262202459600684959330532}{1365220377829828203115595168074635041613268941514172023597} a^{15} - \frac{510707022786953772768524893150405047309188953767226847425}{1365220377829828203115595168074635041613268941514172023597} a^{14} + \frac{387223318185388271085633136514536336441982591853991530596}{1365220377829828203115595168074635041613268941514172023597} a^{13} + \frac{633558940259632069154825855282436871355948815855755359981}{1365220377829828203115595168074635041613268941514172023597} a^{12} - \frac{196836180870030280184716713104369669807561364932637929140}{1365220377829828203115595168074635041613268941514172023597} a^{11} + \frac{657151474320215300207387574591012630778368031730353478772}{1365220377829828203115595168074635041613268941514172023597} a^{10} - \frac{3060420176995503207016226300866757003443141618130264093}{16448438287106363892958977928610060742328541464026168959} a^{9} + \frac{68423659877116820840700539973282086681663002948591630840}{1365220377829828203115595168074635041613268941514172023597} a^{8} - \frac{668893522721584913009825686710269382742777797716759076844}{1365220377829828203115595168074635041613268941514172023597} a^{7} - \frac{525981852067395271039494859393273520073210674999992601111}{1365220377829828203115595168074635041613268941514172023597} a^{6} + \frac{159338131030312929103581753969277716380461935302785354290}{1365220377829828203115595168074635041613268941514172023597} a^{5} - \frac{437679221508859436159799048335632797726365118257863623622}{1365220377829828203115595168074635041613268941514172023597} a^{4} + \frac{501670577804556117310984972579246828435617680984835450807}{1365220377829828203115595168074635041613268941514172023597} a^{3} + \frac{450642196253928842624086071895904658178616299140713306799}{1365220377829828203115595168074635041613268941514172023597} a^{2} + \frac{378745931654771918228547160213929003447539766601602744316}{1365220377829828203115595168074635041613268941514172023597} a - \frac{325941543781279299332382160781623535381532700389688515136}{1365220377829828203115595168074635041613268941514172023597}$
Class group and class number
$C_{53}\times C_{53}$, which has order $2809$ (assuming GRH)
Unit group
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{229987870023954017059236497189737894134743541980}{3137680228602444474088975878125518885635965438371} a^{25} + \frac{217136480836551540404588632151115363363948328202}{3137680228602444474088975878125518885635965438371} a^{24} - \frac{5734368197620153028358493469558145698063203294598}{3137680228602444474088975878125518885635965438371} a^{23} + \frac{2895769853953695503445872483572537182694702551756}{3137680228602444474088975878125518885635965438371} a^{22} - \frac{92903090782795538430449179718308485758584153051060}{3137680228602444474088975878125518885635965438371} a^{21} + \frac{38910900127714861412854367362364439244898433859460}{3137680228602444474088975878125518885635965438371} a^{20} - \frac{825175169192090249025432598882797587021843700983579}{3137680228602444474088975878125518885635965438371} a^{19} + \frac{208619354791426775899170622718184951223205376339973}{3137680228602444474088975878125518885635965438371} a^{18} - \frac{5186025624562473117779830913522560002009908665030605}{3137680228602444474088975878125518885635965438371} a^{17} + \frac{950284046221967557819071670013626109187374876093148}{3137680228602444474088975878125518885635965438371} a^{16} - \frac{20026979216450597144491354405047755726381392519412945}{3137680228602444474088975878125518885635965438371} a^{15} + \frac{26026733398959552629061003677239193487412451502180}{3137680228602444474088975878125518885635965438371} a^{14} - \frac{53640081111470508555505879268919239920346609308326833}{3137680228602444474088975878125518885635965438371} a^{13} + \frac{1493283954000585215753084023395936297151511603120853}{3137680228602444474088975878125518885635965438371} a^{12} - \frac{83693528184162061027106813090510590280448512245797904}{3137680228602444474088975878125518885635965438371} a^{11} + \frac{2190262566064004435436005680569916912366060961235047}{3137680228602444474088975878125518885635965438371} a^{10} - \frac{91866436702243517298271969914984914726430188109122519}{3137680228602444474088975878125518885635965438371} a^{9} + \frac{7761435268729161722573189589323550986192528410571847}{3137680228602444474088975878125518885635965438371} a^{8} - \frac{51689186418254482677926859052944291289597257812419914}{3137680228602444474088975878125518885635965438371} a^{7} + \frac{4755838253545939984437862860116971870622966564403172}{3137680228602444474088975878125518885635965438371} a^{6} - \frac{20044193060199090632308192233670562915612723850599729}{3137680228602444474088975878125518885635965438371} a^{5} + \frac{2415825135601289194505318683440073804435597153774499}{3137680228602444474088975878125518885635965438371} a^{4} - \frac{1267780444146117601025600216051378277239936331305850}{3137680228602444474088975878125518885635965438371} a^{3} - \frac{180775132780258128332551968251888391460806926920950}{3137680228602444474088975878125518885635965438371} a^{2} - \frac{28118311719579089033381263797378756317127766600308}{3137680228602444474088975878125518885635965438371} a + \frac{1646897395182689836224362555359664278287504292319}{3137680228602444474088975878125518885635965438371} \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 5382739421.971964 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 13.13.491258904256726154641.1 |
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 | $26$ | R | $26$ | ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | $26$ | ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | $26$ | R | $26$ |
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 | ||||||
| 53 | Data not computed | ||||||