Normalized defining polynomial
\( x^{26} - 11 x^{25} - 46 x^{24} + 864 x^{23} - 218 x^{22} - 26688 x^{21} + 47802 x^{20} + 420577 x^{19} - 1139464 x^{18} - 3650177 x^{17} + 13015057 x^{16} + 17384092 x^{15} - 84200156 x^{14} - 39841542 x^{13} + 324920368 x^{12} + 9184973 x^{11} - 753477802 x^{10} + 148887155 x^{9} + 1020742598 x^{8} - 291363716 x^{7} - 746387536 x^{6} + 202807416 x^{5} + 242230560 x^{4} - 44893709 x^{3} - 16149195 x^{2} + 3285731 x - 103667 \)
Invariants
| Degree: | $26$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[26, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2390324420009215699399802682569915825064508173649530336497=17^{13}\cdot 53^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $161.01$ | 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: | $17, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(901=17\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{901}(256,·)$, $\chi_{901}(1,·)$, $\chi_{901}(579,·)$, $\chi_{901}(69,·)$, $\chi_{901}(577,·)$, $\chi_{901}(713,·)$, $\chi_{901}(203,·)$, $\chi_{901}(460,·)$, $\chi_{901}(205,·)$, $\chi_{901}(526,·)$, $\chi_{901}(16,·)$, $\chi_{901}(664,·)$, $\chi_{901}(596,·)$, $\chi_{901}(407,·)$, $\chi_{901}(152,·)$, $\chi_{901}(222,·)$, $\chi_{901}(543,·)$, $\chi_{901}(545,·)$, $\chi_{901}(611,·)$, $\chi_{901}(849,·)$, $\chi_{901}(169,·)$, $\chi_{901}(492,·)$, $\chi_{901}(307,·)$, $\chi_{901}(766,·)$, $\chi_{901}(630,·)$, $\chi_{901}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22} - \frac{5}{23} a^{21} - \frac{9}{23} a^{20} - \frac{6}{23} a^{19} + \frac{3}{23} a^{18} + \frac{1}{23} a^{17} + \frac{9}{23} a^{16} - \frac{11}{23} a^{15} + \frac{5}{23} a^{13} - \frac{4}{23} a^{12} - \frac{11}{23} a^{11} - \frac{10}{23} a^{10} + \frac{4}{23} a^{9} + \frac{4}{23} a^{8} + \frac{9}{23} a^{7} - \frac{8}{23} a^{6} + \frac{8}{23} a^{5} - \frac{4}{23} a^{4} + \frac{8}{23} a^{3} - \frac{11}{23} a^{2} + \frac{10}{23} a + \frac{10}{23}$, $\frac{1}{1909} a^{23} + \frac{38}{1909} a^{22} + \frac{190}{1909} a^{21} + \frac{67}{1909} a^{20} - \frac{669}{1909} a^{19} - \frac{261}{1909} a^{18} - \frac{316}{1909} a^{17} + \frac{698}{1909} a^{16} - \frac{565}{1909} a^{15} + \frac{327}{1909} a^{14} + \frac{4}{1909} a^{13} - \frac{597}{1909} a^{12} - \frac{3}{83} a^{11} + \frac{563}{1909} a^{10} + \frac{61}{1909} a^{9} + \frac{20}{1909} a^{8} + \frac{586}{1909} a^{7} + \frac{308}{1909} a^{6} - \frac{189}{1909} a^{5} - \frac{26}{1909} a^{4} + \frac{34}{1909} a^{3} - \frac{164}{1909} a^{2} + \frac{279}{1909} a + \frac{11}{23}$, $\frac{1}{435105007} a^{24} + \frac{31277}{435105007} a^{23} - \frac{2958080}{435105007} a^{22} - \frac{689915}{435105007} a^{21} - \frac{125178196}{435105007} a^{20} - \frac{149214745}{435105007} a^{19} - \frac{155628593}{435105007} a^{18} + \frac{80706493}{435105007} a^{17} + \frac{35336577}{435105007} a^{16} - \frac{21213313}{435105007} a^{15} + \frac{5839729}{435105007} a^{14} - \frac{101204199}{435105007} a^{13} - \frac{2123843}{18917609} a^{12} + \frac{138000532}{435105007} a^{11} + \frac{804003}{18917609} a^{10} + \frac{177363864}{435105007} a^{9} - \frac{6827204}{435105007} a^{8} + \frac{15974892}{435105007} a^{7} - \frac{59543888}{435105007} a^{6} - \frac{120606130}{435105007} a^{5} + \frac{116482341}{435105007} a^{4} + \frac{175962965}{435105007} a^{3} - \frac{80274022}{435105007} a^{2} + \frac{154412}{605153} a - \frac{1412830}{5242229}$, $\frac{1}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{25} - \frac{1443359108706847187476068107364330592158041567606948296345703195064287545591}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{24} - \frac{1114036498853300356338541511376312704395583927318196184724305033228023275880730938}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{23} + \frac{39473999036827537414721243484646548815993195336989175796118527847717577160808741575}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{22} - \frac{761358552996428581416804383634446272615323702436165843328268008011859907247583943200}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{21} - \frac{2349147193615925209466583814269731290018440432204556539392235477372537362656594393913}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{20} + \frac{990966900381535241058897952905214926483723004989436158850090997354882193402266276075}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{19} + \frac{1394543311789127604106290759737414268565977533827807285715114505351925810215269761472}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{18} - \frac{658826891760122038697784490127479852509048444922993456222286412162411566696925781619}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{17} - \frac{1037994361288235380740690505012007048168139981203947130986330492948779148890694404278}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{16} - \frac{2301123011758029824752958432896160773375031033629860561676683995490973608758138898184}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{15} - \frac{2137953986476138962220633154973831984130311289383469376516099542309500376109672804289}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{14} + \frac{5979028756378739438236788015549603051549316440848494822031728747740203744445509870}{205706263321569394862757752381768958227424741890483490718990471196207070953542349893} a^{13} + \frac{700656195766062616761025421308132347911103669465968244848620087910391968099314822568}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{12} - \frac{532562258714477345161557766684984509301097083961594203853676169037875087079769519}{2478388714717703552563346414238180219607527010728716755650487604773579168114968071} a^{11} + \frac{1790100862865195008944833777451914107790993259609162866706338932087370399965655098754}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{10} - \frac{2001740549795655309767068487419743364683707541843577786564793589918579279960480277080}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{9} - \frac{265865110510621329814774380753707680427865347295611789297215588610050112722016430654}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{8} + \frac{627355518954022678536686808528411843030136424510172897023643180703220223346380295895}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{7} + \frac{930448561431204309865912569392173687305813851623646954298146522203296511658652575280}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{6} + \frac{101573715390819629490802229184473674833995953422888313408329669779799798140775166657}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{5} - \frac{1941571354089978578687119988943218144249348896759933169111189880977286780012468201959}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{4} + \frac{2057398290433373782084449317276891380776973747934972449368138515575324489873050024494}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{3} + \frac{2204589158320063231523884577184965396389534555472492280260269239529663672486394870660}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a^{2} - \frac{1866513110987670871049407654915223051367812477321305713862014100512598623176013404608}{4731244056396096081843428304780686039230769063481120286536780837512762631931474047539} a + \frac{1236260261118668251953828086011112770674994111079186867406744945374275494681418886}{2478388714717703552563346414238180219607527010728716755650487604773579168114968071}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $25$ | 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: | \( 189533419563539170000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
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
| \(\Q(\sqrt{17}) \), 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 | ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | $26$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | R | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | $26$ | $26$ | $26$ | $26$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ | R | ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$ |
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 | ||||||
| $53$ | 53.13.12.1 | $x^{13} - 53$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |
| 53.13.12.1 | $x^{13} - 53$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ | |