Normalized defining polynomial
\( x^{21} - 9 x^{20} - 166 x^{19} + 1681 x^{18} + 10386 x^{17} - 132026 x^{16} - 246888 x^{15} + 5573674 x^{14} - 2917633 x^{13} - 131793916 x^{12} + 307937724 x^{11} + 1544609763 x^{10} - 7129049062 x^{9} - 2245737177 x^{8} + 67402052396 x^{7} - 128090775669 x^{6} - 103888012885 x^{5} + 811138023075 x^{4} - 1505292777223 x^{3} + 1441214825042 x^{2} - 731120240340 x + 156367753544 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[21, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2402550119368274484118955435600030173163169467760129642496=2^{12}\cdot 73^{2}\cdot 149^{6}\cdot 211^{6}\cdot 1009^{2}\cdot 334607797^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $540.02$ | 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: | $2, 73, 149, 211, 1009, 334607797$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{2} a^{17} + \frac{1}{4} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{30265947092233553479345288553798086404158403892978528} a^{20} - \frac{3452477459171175897960801922414604197140239620615491}{30265947092233553479345288553798086404158403892978528} a^{19} - \frac{428680638405476391291725494496696772115693760092073}{3783243386529194184918161069224760800519800486622316} a^{18} + \frac{7854770703320497679611022801182914828068556815372161}{30265947092233553479345288553798086404158403892978528} a^{17} + \frac{1783491133964975334314471298350105513612578731747951}{3783243386529194184918161069224760800519800486622316} a^{16} - \frac{5499372860610651354944084811679577902355305584275981}{15132973546116776739672644276899043202079201946489264} a^{15} + \frac{3451191216003056466483776015111415145740383223096683}{7566486773058388369836322138449521601039600973244632} a^{14} - \frac{652953144772866054502816368842593721205247330295695}{15132973546116776739672644276899043202079201946489264} a^{13} + \frac{12399219511527673751341422279501121800608368034634171}{30265947092233553479345288553798086404158403892978528} a^{12} - \frac{6784881231734063984773696466766351154223044996364517}{15132973546116776739672644276899043202079201946489264} a^{11} + \frac{72141580967195666901841165655529615180570220568640}{945810846632298546229540267306190200129950121655579} a^{10} - \frac{1349509156279763454222633885611350286509416541853053}{30265947092233553479345288553798086404158403892978528} a^{9} + \frac{130154153639149706191880067929441915908937649271307}{7566486773058388369836322138449521601039600973244632} a^{8} - \frac{2184713307091666709050826943093730054975716081345185}{30265947092233553479345288553798086404158403892978528} a^{7} + \frac{4290532479549399459807129244977955148073570170709803}{15132973546116776739672644276899043202079201946489264} a^{6} + \frac{5914994759358551891749339404443303545009957841096751}{30265947092233553479345288553798086404158403892978528} a^{5} + \frac{12371320447459454131524012293195473968010452454379813}{30265947092233553479345288553798086404158403892978528} a^{4} + \frac{13068081638822896319935674405201447057855033296369073}{30265947092233553479345288553798086404158403892978528} a^{3} + \frac{7917973563462241224710358698748285050048056090340895}{30265947092233553479345288553798086404158403892978528} a^{2} + \frac{2331131912921125895490827589358138386047178674630535}{7566486773058388369836322138449521601039600973244632} a + \frac{3634717098419724724868474063712711488384667987188405}{7566486773058388369836322138449521601039600973244632}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $20$ | 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: | \( 35402389691000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 352719360 |
| The 150 conjugacy class representatives for t21n148 are not computed |
| Character table for t21n148 is not computed |
Intermediate fields
| 7.7.988410721.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 sibling: | 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 | R | $21$ | $21$ | $21$ | $15{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | $21$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.10.14 | $x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
| 73 | Data not computed | ||||||
| 149 | Data not computed | ||||||
| 211 | Data not computed | ||||||
| 1009 | Data not computed | ||||||
| 334607797 | Data not computed | ||||||