Normalized defining polynomial
\( x^{21} - 4 x^{20} - 154 x^{19} + 792 x^{18} + 7952 x^{17} - 51926 x^{16} - 145220 x^{15} + 1418338 x^{14} + 61944 x^{13} - 16336335 x^{12} + 18143395 x^{11} + 79537118 x^{10} - 132545343 x^{9} - 176540462 x^{8} + 368567997 x^{7} + 170444833 x^{6} - 445036167 x^{5} - 52948234 x^{4} + 208339119 x^{3} - 3670640 x^{2} - 25278358 x + 1626227 \)
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: | \(6120466749549222213884941343724120599616956835121=7^{14}\cdot 113^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $210.47$ | 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: | $7, 113$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(791=7\cdot 113\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{791}(256,·)$, $\chi_{791}(1,·)$, $\chi_{791}(708,·)$, $\chi_{791}(141,·)$, $\chi_{791}(16,·)$, $\chi_{791}(275,·)$, $\chi_{791}(340,·)$, $\chi_{791}(674,·)$, $\chi_{791}(219,·)$, $\chi_{791}(30,·)$, $\chi_{791}(480,·)$, $\chi_{791}(162,·)$, $\chi_{791}(242,·)$, $\chi_{791}(106,·)$, $\chi_{791}(109,·)$, $\chi_{791}(561,·)$, $\chi_{791}(114,·)$, $\chi_{791}(501,·)$, $\chi_{791}(694,·)$, $\chi_{791}(445,·)$, $\chi_{791}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11461} a^{18} + \frac{4465}{11461} a^{17} + \frac{1853}{11461} a^{16} + \frac{5712}{11461} a^{15} - \frac{3147}{11461} a^{14} + \frac{1006}{11461} a^{13} + \frac{1938}{11461} a^{12} + \frac{1899}{11461} a^{11} + \frac{3053}{11461} a^{10} + \frac{1881}{11461} a^{9} + \frac{4206}{11461} a^{8} - \frac{41}{157} a^{7} + \frac{1594}{11461} a^{6} + \frac{4421}{11461} a^{5} + \frac{5059}{11461} a^{4} + \frac{2269}{11461} a^{3} - \frac{3539}{11461} a^{2} - \frac{64}{11461} a - \frac{2137}{11461}$, $\frac{1}{11461} a^{19} - \frac{3693}{11461} a^{17} - \frac{4552}{11461} a^{16} + \frac{4959}{11461} a^{15} + \frac{1175}{11461} a^{14} + \frac{2860}{11461} a^{13} + \frac{1784}{11461} a^{12} + \frac{5158}{11461} a^{11} - \frac{2635}{11461} a^{10} - \frac{5007}{11461} a^{9} + \frac{1796}{11461} a^{8} + \frac{1813}{11461} a^{7} + \frac{4492}{11461} a^{6} + \frac{1136}{11461} a^{5} + \frac{3465}{11461} a^{4} - \frac{3100}{11461} a^{3} - \frac{3148}{11461} a^{2} - \frac{2902}{11461} a - \frac{5308}{11461}$, $\frac{1}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{20} - \frac{6037327589845061997867535383739538486433916183117815888237635248762437}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{19} + \frac{196673616103622238538164811100981406097540241270409110343511959869156483}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{18} - \frac{2273122275889291841118995704844747254671682937890232495589775728257025722014}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{17} + \frac{1432094875715837450217524529852732558543085743579674596075526733351091961317}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{16} - \frac{624839482337881697442305708505444681353811226804360662883640323531960686338}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{15} - \frac{284614346563293770983624520307790119009047092442683817682498053942665513673}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{14} - \frac{895803626521815304328848478208357059391903338492786939379805863459935548297}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{13} - \frac{791535487902920207714081149101032051560581914145921235454802753545115595}{3652938187416791243177637289769478715901632151897596519915907779651346071} a^{12} + \frac{1891237267877341259076027712425002581799379854283894479761140805666979518342}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{11} + \frac{2647358339746681071227584100772750139798767450583285398458367759579008836647}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{10} - \frac{1504153376171812247968774654798129820707929729779097235936822332656670985962}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{9} + \frac{476288716608861900363322338349126027538117879027902958519356697033618161771}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{8} + \frac{1172034054077691259786192556169289163044855396814257857717454567602392363939}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{7} + \frac{2295682240005039558003651934233416908196246704067957075596716130433116770875}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{6} + \frac{26710503520058409033251690533046811596221491999097909248064403828358502150}{78413070406604272302182981274914700655039144959226489680934623160461086209} a^{5} - \frac{2139499642816029177909254565983755620633572417733335032995886422179409362934}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{4} - \frac{14642665757016624416944861254430540394056625109922229570173662895644109343}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{3} + \frac{602191377951273683073165683059009222805417027835423457139030845201737596728}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a^{2} + \frac{1775391190492677816649439148912312019466737712254162523820268359199049052478}{5724154139682111878059357633068773147817857582023533746708227490713659293257} a + \frac{846762154884790934626739454422323209710709572725866093368280572431789330155}{5724154139682111878059357633068773147817857582023533746708227490713659293257}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 350292924482315400 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.7.2081951752609.1 |
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 | $21$ | $21$ | $21$ | R | $21$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ |
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 | Data not computed | ||||||
| $113$ | 113.7.6.1 | $x^{7} - 113$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 113.7.6.1 | $x^{7} - 113$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 113.7.6.1 | $x^{7} - 113$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |