Normalized defining polynomial
\( x^{21} - 8 x^{20} - 44 x^{19} + 177 x^{18} + 930 x^{17} + 13477 x^{16} - 123876 x^{15} - 187878 x^{14} + 6288822 x^{13} - 11318947 x^{12} - 109350935 x^{11} + 301750779 x^{10} + 715568742 x^{9} - 1607781301 x^{8} + 156769574 x^{7} - 6541085155 x^{6} - 34174260492 x^{5} - 24325229292 x^{4} + 54837372042 x^{3} - 11194152858 x^{2} - 181242408040 x + 76054217719 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[9, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(669989641990779134661554841839710781441002093=7^{4}\cdot 13^{7}\cdot 109^{6}\cdot 307^{2}\cdot 167732941^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $136.32$ | 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: | $7, 13, 109, 307, 167732941$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{20} + \frac{16122681044645270481250011388841293510830787240673134597302718789029458676828551892625563032063941760468810595374689475775942369}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{19} + \frac{697175963980390072547655184805922431276953519052693013305480144116044188917532140208831558586916181095222899451876435140895970}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{18} + \frac{14815872270505024078060557504270718035271921403021477374837078632291691689821517528695663765649878833530237358931323194111407831}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{17} - \frac{4615747416719745065383176995770016383530233737800474016171189680508070646232208006604781566842271733435731814074268792845478395}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{16} + \frac{6496030751361721451525991944952129572217312866130004024763828822559028613145796405290460168861703768581420079822177120308710389}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{15} - \frac{16649304434797842498623889387445288516534358014806537861694605356833946125174951268854324819315538985177661255462389161512100416}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{14} - \frac{11668610974978186414217723325687964321408366550990871234658928920517853620319999156435906216034422697620140419696689965514939232}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{13} + \frac{5151650851079985905200227673336009505315035156653816657049314381998077405141223207464566648462333883643275374430337951687175526}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{12} + \frac{16330157415115489020373846990126530666796032950742661730888517653706651200279836256818396183860486010367537516908229796001398301}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{11} + \frac{11896448241102953348135312812545955525178577387970703124875338655644557075083254758795230208727422628202682005520195613029464560}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{10} + \frac{4171817067763312498364512411036572703971229451464058587755703839962012068092281139187766934580119454945482472796780459649934309}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{9} - \frac{19096566164460251945378092507211810174428022566040384632123835960315665807891143239923112072107804555232597691456272390853385868}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{8} + \frac{510713171404719888627696959579174968944905750970024407629255302153094705250872550150014502316475336522293136276011300584243056}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{7} + \frac{17383611023055867685270955468009419073213877357452750646459378324306856863662807933200130035126199857271049970682164259371627644}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{6} + \frac{17162973998044272365176159643701631982887659291461317358977981085204968076009913652218436009231855495054415661870869891857207523}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{5} - \frac{6289303654086708706850572258965518044620216119610450463119172947734916029539798908098946861822602043965652373745179925839360121}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{4} - \frac{2882740948923509893164031953627330688339562342015190804129687401705714777648465719629121633441385480525839449792784871043896088}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{3} + \frac{11125548565231588899625176749967839922392914628541543971747912793993368116158038144748272477417299069027077448799834042543809907}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a^{2} + \frac{13708555497571395829788576379753748846441319774700933553460943972003804759880127067651100430964173155493226858139632721417631342}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759} a - \frac{15218313892013811977160215592516042424874495908653495280734136157509645482135638540267499850451790517533386396380489720683883997}{38697529147045868424814865398833874840171462890907116371264641888978419021359354402593529789258897267114127864075688132858889759}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 456445190009000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 5878656 |
| The 120 conjugacy class representatives for t21n136 are not computed |
| Character table for t21n136 is not computed |
Intermediate fields
| 7.3.2007889.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ | ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | $21$ | ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $21$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.6.3.2 | $x^{6} - 49 x^{2} + 686$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $109$ | $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.4.2.1 | $x^{4} + 1199 x^{2} + 427716$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 109.4.2.1 | $x^{4} + 1199 x^{2} + 427716$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 109.4.2.1 | $x^{4} + 1199 x^{2} + 427716$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 307 | Data not computed | ||||||
| 167732941 | Data not computed | ||||||