Normalized defining polynomial
\( x^{21} - 4 x^{20} - 241 x^{19} + 372 x^{18} + 23007 x^{17} + 15672 x^{16} - 1077907 x^{15} - 2967733 x^{14} + 23479824 x^{13} + 127554499 x^{12} - 92652808 x^{11} - 2223887778 x^{10} - 4880010884 x^{9} + 10799855153 x^{8} + 70086425352 x^{7} + 99635117252 x^{6} - 134450275449 x^{5} - 739782266037 x^{4} - 1366706473054 x^{3} - 1469609407945 x^{2} - 908380706592 x - 234723929868 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[13, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(17506169099840522133726465527996746208402356008227614249=7^{14}\cdot 31^{2}\cdot 79^{2}\cdot 151^{2}\cdot 173^{10}\cdot 88657^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $427.20$ | 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: | $7, 31, 79, 151, 173, 88657$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7} a^{19} + \frac{1}{7} a^{18} + \frac{1}{7} a^{17} - \frac{2}{7} a^{16} + \frac{1}{7} a^{15} - \frac{1}{7} a^{14} + \frac{3}{7} a^{13} + \frac{3}{7} a^{12} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{3039168907178728151846012188048355964728622683249108455373178237739824446262219067151105452176081599762966791284} a^{20} - \frac{45291112483829551128945486964672961217356169868492983833597705869663811934349420480297620308443818316243285217}{1519584453589364075923006094024177982364311341624554227686589118869912223131109533575552726088040799881483395642} a^{19} - \frac{298783774222734017941710773979825427894022300119138523639753508315514511406609658183697020100511762096663004777}{3039168907178728151846012188048355964728622683249108455373178237739824446262219067151105452176081599762966791284} a^{18} - \frac{180268736048976156782035016808750721899955261032764151905869409725078445300742524013307627550055107491813924401}{506528151196454691974335364674725994121437113874851409228863039623304074377036511191850908696013599960494465214} a^{17} - \frac{214350081868205349941420916812619300054740312060675742092188391303341568498285628663159225185718710022191074039}{1013056302392909383948670729349451988242874227749702818457726079246608148754073022383701817392027199920988930428} a^{16} - \frac{2139077354566569906012549775449113822987894709640982131920807275637891670074639794580537103498635213322645979}{72361164456636384567762194953532284874491016267835915604123291374757724911005215884550129813716228565784923602} a^{15} - \frac{72906311340613381219252218357490493877078647594492062083586644522750376571115658320768731345332487504991926611}{3039168907178728151846012188048355964728622683249108455373178237739824446262219067151105452176081599762966791284} a^{14} - \frac{927210356080950564246941832847864441406408901030795301688602372970634840061477827196582212918003329830745643019}{3039168907178728151846012188048355964728622683249108455373178237739824446262219067151105452176081599762966791284} a^{13} + \frac{190771849327205055669847427156269719738924678432896683473399267818420139085568833476430353176207363335577982487}{506528151196454691974335364674725994121437113874851409228863039623304074377036511191850908696013599960494465214} a^{12} - \frac{800146418418430050196162208523677240421840503551664103209933574394536893456748188570508034139212634718548811149}{3039168907178728151846012188048355964728622683249108455373178237739824446262219067151105452176081599762966791284} a^{11} + \frac{5367860189567925119694554217909130544530472389068330672913851673489754402773488083656764177821730807325631977}{217083493369909153703286584860596854623473048803507746812369874124273174733015647653650389441148685697354770806} a^{10} - \frac{8672797627468263165329829923265619005064169223748960704570592339547648867275223093604065799522544341003417431}{72361164456636384567762194953532284874491016267835915604123291374757724911005215884550129813716228565784923602} a^{9} - \frac{8015011239582514009016845566644714678689892957982833880786260334138236147669306808268838765658566897677367291}{108541746684954576851643292430298427311736524401753873406184937062136587366507823826825194720574342848677385403} a^{8} + \frac{14569358450094865770912818593170867405807875669181582783321352430151026642774686095564993354781846009280922869}{3039168907178728151846012188048355964728622683249108455373178237739824446262219067151105452176081599762966791284} a^{7} + \frac{127040973484685273562819860254125681033651321741912872256052853865933136928596061273473892955320011038162803257}{506528151196454691974335364674725994121437113874851409228863039623304074377036511191850908696013599960494465214} a^{6} + \frac{375724381281507968258771123122099997373978381584905042954265492053697831622733951259971233245502146452622974022}{759792226794682037961503047012088991182155670812277113843294559434956111565554766787776363044020399940741697821} a^{5} - \frac{220364821029333410692565490794537112760211083536072740529403995239336579566475652700523873715225491999921396651}{1013056302392909383948670729349451988242874227749702818457726079246608148754073022383701817392027199920988930428} a^{4} - \frac{54207359957875518895522524559924222004832102160891422713343272560139698411806640901638861774176296583978788867}{144722328913272769135524389907064569748982032535671831208246582749515449822010431769100259627432457131569847204} a^{3} - \frac{26976285089841267823107237211562217055899308321952349155834987116025678152137563828311773237644185246761173139}{759792226794682037961503047012088991182155670812277113843294559434956111565554766787776363044020399940741697821} a^{2} + \frac{1385961356823335053562979274758739481943659197327605597843343538716213742131278040473754251089709619605308134859}{3039168907178728151846012188048355964728622683249108455373178237739824446262219067151105452176081599762966791284} a - \frac{113843025575619996080552560275964891302272736365540930131444546622306423123193228636556588337644803154588565597}{506528151196454691974335364674725994121437113874851409228863039623304074377036511191850908696013599960494465214}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $16$ | 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: | \( 556089488945000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 5878656 |
| The 96 conjugacy class representatives for t21n134 are not computed |
| Character table for t21n134 is not computed |
Intermediate fields
| 7.7.12431698517.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $21$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.12.10.5 | $x^{12} + 56 x^{6} + 1323$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ | |
| $31$ | 31.3.0.1 | $x^{3} - x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 31.3.2.2 | $x^{3} + 217$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.0.1 | $x^{3} - x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 31.6.0.1 | $x^{6} - 2 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 31.6.0.1 | $x^{6} - 2 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 79 | Data not computed | ||||||
| 151 | Data not computed | ||||||
| 173 | Data not computed | ||||||
| 88657 | Data not computed | ||||||