Normalized defining polynomial
\( x^{21} - x^{20} - x^{19} + 2 x^{18} + 5 x^{17} + 2 x^{16} - 10 x^{15} - 2 x^{14} + 23 x^{13} + 19 x^{12} - 11 x^{11} - 28 x^{10} + 8 x^{9} + 47 x^{8} + 23 x^{7} - 13 x^{6} - 24 x^{5} - 7 x^{4} + 11 x^{3} + \cdots - 1 \)
Invariants
Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(156106141952973043121291857\) \(\medspace = 23^{7}\cdot 71^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(17.67\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $23^{1/2}71^{1/2}\approx 40.4103947023535$ | ||
Ramified primes: | \(23\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1633}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7}a^{17}+\frac{1}{7}a^{16}-\frac{3}{7}a^{15}-\frac{1}{7}a^{14}-\frac{3}{7}a^{13}+\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}+\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}-\frac{3}{7}a^{4}-\frac{2}{7}$, $\frac{1}{7}a^{18}+\frac{3}{7}a^{16}+\frac{2}{7}a^{15}-\frac{2}{7}a^{14}-\frac{1}{7}a^{13}+\frac{3}{7}a^{12}+\frac{3}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{7}a^{8}+\frac{3}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{19}-\frac{1}{7}a^{16}+\frac{2}{7}a^{14}-\frac{2}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}+\frac{1}{7}a^{10}+\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{4}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{67050907}a^{20}+\frac{619012}{9578701}a^{19}-\frac{2430104}{67050907}a^{18}-\frac{48420}{9578701}a^{17}+\frac{1154233}{9578701}a^{16}+\frac{25935584}{67050907}a^{15}+\frac{2144834}{9578701}a^{14}-\frac{21230934}{67050907}a^{13}+\frac{21553102}{67050907}a^{12}-\frac{1793716}{67050907}a^{11}+\frac{25671549}{67050907}a^{10}-\frac{19772818}{67050907}a^{9}+\frac{31783945}{67050907}a^{8}-\frac{1214318}{9578701}a^{7}-\frac{17854070}{67050907}a^{6}+\frac{28224926}{67050907}a^{5}-\frac{23307932}{67050907}a^{4}+\frac{24879307}{67050907}a^{3}-\frac{28098866}{67050907}a^{2}+\frac{7421087}{67050907}a-\frac{6164758}{67050907}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{12008369}{67050907}a^{20}-\frac{33047792}{67050907}a^{19}+\frac{22843134}{67050907}a^{18}+\frac{26725256}{67050907}a^{17}+\frac{10752985}{67050907}a^{16}-\frac{53248837}{67050907}a^{15}-\frac{97689134}{67050907}a^{14}+\frac{191647202}{67050907}a^{13}+\frac{24536663}{9578701}a^{12}-\frac{227716730}{67050907}a^{11}-\frac{219135061}{67050907}a^{10}+\frac{9267503}{9578701}a^{9}+\frac{448525808}{67050907}a^{8}+\frac{72959049}{67050907}a^{7}-\frac{447223565}{67050907}a^{6}-\frac{73499782}{67050907}a^{5}+\frac{56067461}{67050907}a^{4}+\frac{166836778}{67050907}a^{3}+\frac{49747284}{67050907}a^{2}-\frac{140417181}{67050907}a-\frac{14278335}{67050907}$, $\frac{8887528}{67050907}a^{20}-\frac{5276769}{67050907}a^{19}-\frac{421194}{9578701}a^{18}-\frac{113737}{67050907}a^{17}+\frac{55672847}{67050907}a^{16}+\frac{46271614}{67050907}a^{15}-\frac{44987521}{67050907}a^{14}-\frac{6257727}{9578701}a^{13}+\frac{132251481}{67050907}a^{12}+\frac{274580767}{67050907}a^{11}+\frac{123352825}{67050907}a^{10}-\frac{30454823}{9578701}a^{9}-\frac{38174746}{67050907}a^{8}+\frac{46769497}{9578701}a^{7}+\frac{451246056}{67050907}a^{6}+\frac{192928256}{67050907}a^{5}-\frac{144150016}{67050907}a^{4}-\frac{48805293}{67050907}a^{3}+\frac{2342726}{9578701}a^{2}+\frac{14897336}{67050907}a+\frac{49451407}{67050907}$, $\frac{22489672}{67050907}a^{20}-\frac{33432431}{67050907}a^{19}+\frac{4118413}{67050907}a^{18}+\frac{4704945}{9578701}a^{17}+\frac{13135473}{9578701}a^{16}+\frac{11894625}{67050907}a^{15}-\frac{178824222}{67050907}a^{14}+\frac{75382215}{67050907}a^{13}+\frac{57463174}{9578701}a^{12}+\frac{28982315}{9578701}a^{11}-\frac{154429789}{67050907}a^{10}-\frac{325116961}{67050907}a^{9}+\frac{338096482}{67050907}a^{8}+\frac{617147831}{67050907}a^{7}+\frac{214770755}{67050907}a^{6}+\frac{5561692}{9578701}a^{5}-\frac{201381921}{67050907}a^{4}-\frac{94581715}{67050907}a^{3}-\frac{8593195}{67050907}a^{2}+\frac{20566820}{67050907}a+\frac{64334063}{67050907}$, $\frac{7278451}{67050907}a^{20}-\frac{20830646}{67050907}a^{19}+\frac{13762131}{67050907}a^{18}+\frac{17103208}{67050907}a^{17}+\frac{8773211}{67050907}a^{16}-\frac{41217677}{67050907}a^{15}-\frac{62917751}{67050907}a^{14}+\frac{119394070}{67050907}a^{13}+\frac{118913021}{67050907}a^{12}-\frac{145171162}{67050907}a^{11}-\frac{175926347}{67050907}a^{10}+\frac{41484117}{67050907}a^{9}+\frac{302900347}{67050907}a^{8}+\frac{9138292}{9578701}a^{7}-\frac{277482844}{67050907}a^{6}-\frac{65597912}{67050907}a^{5}+\frac{31243986}{67050907}a^{4}+\frac{108651232}{67050907}a^{3}+\frac{33783481}{67050907}a^{2}+\frac{939566}{67050907}a+\frac{22369884}{67050907}$, $\frac{16040933}{67050907}a^{20}-\frac{41707214}{67050907}a^{19}+\frac{10888033}{67050907}a^{18}+\frac{6152127}{9578701}a^{17}+\frac{33238605}{67050907}a^{16}-\frac{75185357}{67050907}a^{15}-\frac{32488419}{9578701}a^{14}+\frac{153244692}{67050907}a^{13}+\frac{338544386}{67050907}a^{12}-\frac{26362744}{9578701}a^{11}-\frac{565116098}{67050907}a^{10}-\frac{430299269}{67050907}a^{9}+\frac{66111854}{9578701}a^{8}+\frac{477109361}{67050907}a^{7}-\frac{520337718}{67050907}a^{6}-\frac{713627975}{67050907}a^{5}-\frac{616159086}{67050907}a^{4}-\frac{37045104}{67050907}a^{3}+\frac{204717496}{67050907}a^{2}-\frac{15170024}{67050907}a+\frac{46714978}{67050907}$, $\frac{2704416}{6095537}a^{20}-\frac{4418462}{6095537}a^{19}-\frac{944630}{6095537}a^{18}+\frac{6870274}{6095537}a^{17}+\frac{10995805}{6095537}a^{16}-\frac{4455616}{6095537}a^{15}-\frac{29238339}{6095537}a^{14}+\frac{11090250}{6095537}a^{13}+\frac{66673450}{6095537}a^{12}+\frac{13421304}{6095537}a^{11}-\frac{64207681}{6095537}a^{10}-\frac{55510962}{6095537}a^{9}+\frac{73380672}{6095537}a^{8}+\frac{113457210}{6095537}a^{7}-\frac{13255122}{6095537}a^{6}-\frac{77185698}{6095537}a^{5}-\frac{39086402}{6095537}a^{4}+\frac{26062517}{6095537}a^{3}+\frac{6373848}{870791}a^{2}+\frac{1071640}{6095537}a-\frac{10180248}{6095537}$, $\frac{6140432}{6095537}a^{20}-\frac{8740339}{6095537}a^{19}-\frac{2468645}{6095537}a^{18}+\frac{13423290}{6095537}a^{17}+\frac{24458515}{6095537}a^{16}+\frac{2470403}{6095537}a^{15}-\frac{61792193}{6095537}a^{14}+\frac{12012998}{6095537}a^{13}+\frac{135044531}{6095537}a^{12}+\frac{59344066}{6095537}a^{11}-\frac{88736094}{6095537}a^{10}-\frac{135617077}{6095537}a^{9}+\frac{95168233}{6095537}a^{8}+\frac{247718587}{6095537}a^{7}+\frac{42957654}{6095537}a^{6}-\frac{97123202}{6095537}a^{5}-\frac{112422672}{6095537}a^{4}-\frac{7998726}{6095537}a^{3}+\frac{71192463}{6095537}a^{2}+\frac{6805847}{6095537}a-\frac{4177671}{6095537}$, $\frac{3828322}{9578701}a^{20}-\frac{783098}{67050907}a^{19}-\frac{50059839}{67050907}a^{18}+\frac{31748593}{67050907}a^{17}+\frac{172749974}{67050907}a^{16}+\frac{27007069}{9578701}a^{15}-\frac{194958361}{67050907}a^{14}-\frac{39187079}{9578701}a^{13}+\frac{77224518}{9578701}a^{12}+\frac{1027922908}{67050907}a^{11}+\frac{37166927}{9578701}a^{10}-\frac{846234808}{67050907}a^{9}-\frac{442419401}{67050907}a^{8}+\frac{1290723344}{67050907}a^{7}+\frac{1650909879}{67050907}a^{6}+\frac{414358501}{67050907}a^{5}-\frac{642490883}{67050907}a^{4}-\frac{102421676}{9578701}a^{3}-\frac{40422499}{67050907}a^{2}+\frac{199562404}{67050907}a+\frac{72863269}{67050907}$, $\frac{50258345}{67050907}a^{20}-\frac{52389220}{67050907}a^{19}-\frac{46321008}{67050907}a^{18}+\frac{102617090}{67050907}a^{17}+\frac{236851423}{67050907}a^{16}+\frac{97357357}{67050907}a^{15}-\frac{486961026}{67050907}a^{14}-\frac{81113764}{67050907}a^{13}+\frac{1111087753}{67050907}a^{12}+\frac{871424593}{67050907}a^{11}-\frac{499848119}{67050907}a^{10}-\frac{1294404238}{67050907}a^{9}+\frac{327945747}{67050907}a^{8}+\frac{2141519764}{67050907}a^{7}+\frac{1077104519}{67050907}a^{6}-\frac{460370726}{67050907}a^{5}-\frac{1098107610}{67050907}a^{4}-\frac{546448211}{67050907}a^{3}+\frac{329666790}{67050907}a^{2}+\frac{248339249}{67050907}a+\frac{59141074}{67050907}$, $\frac{2705595}{3944171}a^{20}-\frac{4028547}{3944171}a^{19}-\frac{520180}{3944171}a^{18}+\frac{6235961}{3944171}a^{17}+\frac{9402754}{3944171}a^{16}+\frac{638140}{3944171}a^{15}-\frac{24547823}{3944171}a^{14}+\frac{1492892}{563453}a^{13}+\frac{56010700}{3944171}a^{12}+\frac{17008528}{3944171}a^{11}-\frac{32749174}{3944171}a^{10}-\frac{38609041}{3944171}a^{9}+\frac{48598562}{3944171}a^{8}+\frac{89856903}{3944171}a^{7}+\frac{1156360}{563453}a^{6}-\frac{2858385}{563453}a^{5}-\frac{21965931}{3944171}a^{4}+\frac{149351}{3944171}a^{3}+\frac{23578965}{3944171}a^{2}-\frac{1889574}{3944171}a-\frac{4405369}{3944171}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 32527.0761402 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 32527.0761402 \cdot 1}{2\cdot\sqrt{156106141952973043121291857}}\cr\approx \mathstrut & 0.249651232568 \end{aligned}\] (assuming GRH)
Galois group
$S_3\times D_7$ (as 21T8):
A solvable group of order 84 |
The 15 conjugacy class representatives for $S_3\times D_7$ |
Character table for $S_3\times D_7$ |
Intermediate fields
3.1.23.1, 7.1.357911.1 |
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 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $21$ | $21$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.2.0.1}{2} }^{10}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{10}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.2.0.1}{2} }^{10}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | R | $21$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.2.0.1}{2} }^{10}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(71\) | 71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |