Normalized defining polynomial
\( x^{22} - 11 x^{21} + 41 x^{20} - 25 x^{19} - 223 x^{18} + 525 x^{17} + 92 x^{16} - 1654 x^{15} + 1480 x^{14} + 1772 x^{13} - 3759 x^{12} + 506 x^{11} + 3901 x^{10} - 2800 x^{9} - 1867 x^{8} + \cdots - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[20, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-15891297445653813609959315666834999\) \(\medspace = -\,79\cdot 1451^{2}\cdot 18521\cdot 71823490019^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.86\) | 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))
| |
Galois root discriminant: | $79^{1/2}1451^{1/2}18521^{1/2}71823490019^{1/2}\approx 12348457743.510412$ | ||
Ramified primes: | \(79\), \(1451\), \(18521\), \(71823490019\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1463159}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7}a^{18}-\frac{2}{7}a^{17}+\frac{1}{7}a^{15}-\frac{2}{7}a^{14}-\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{9}+\frac{2}{7}a^{8}-\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{2}+\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{19}+\frac{3}{7}a^{17}+\frac{1}{7}a^{16}+\frac{3}{7}a^{14}-\frac{1}{7}a^{13}+\frac{3}{7}a^{12}+\frac{3}{7}a^{11}-\frac{3}{7}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{20}-\frac{2}{7}a^{14}+\frac{3}{7}a^{13}-\frac{1}{7}a^{12}+\frac{3}{7}a^{11}+\frac{3}{7}a^{10}+\frac{3}{7}a^{9}-\frac{2}{7}a^{8}+\frac{3}{7}a^{7}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{21}-\frac{2}{7}a^{15}+\frac{3}{7}a^{14}-\frac{1}{7}a^{13}+\frac{3}{7}a^{12}+\frac{3}{7}a^{11}+\frac{3}{7}a^{10}-\frac{2}{7}a^{9}+\frac{3}{7}a^{8}-\frac{3}{7}a^{6}+\frac{2}{7}a^{5}+\frac{2}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $a^{20}-10a^{19}+32a^{18}-3a^{17}-194a^{16}+328a^{15}+226a^{14}-1100a^{13}+606a^{12}+1278a^{11}-1875a^{10}-91a^{9}+1935a^{8}-956a^{7}-888a^{6}+721a^{5}+168a^{4}-177a^{3}-13a^{2}+12a+2$, $2a^{20}-20a^{19}+\frac{444}{7}a^{18}-\frac{6}{7}a^{17}-401a^{16}+\frac{4504}{7}a^{15}+\frac{3879}{7}a^{14}-2287a^{13}+\frac{6864}{7}a^{12}+\frac{20917}{7}a^{11}-3680a^{10}-\frac{6309}{7}a^{9}+\frac{30372}{7}a^{8}-\frac{10061}{7}a^{7}-\frac{17617}{7}a^{6}+\frac{9876}{7}a^{5}+\frac{5342}{7}a^{4}-430a^{3}-\frac{754}{7}a^{2}+\frac{261}{7}a+\frac{27}{7}$, $\frac{16}{7}a^{20}-\frac{160}{7}a^{19}+\frac{498}{7}a^{18}+\frac{78}{7}a^{17}-\frac{3436}{7}a^{16}+\frac{5048}{7}a^{15}+\frac{5891}{7}a^{14}-\frac{20029}{7}a^{13}+809a^{12}+4285a^{11}-\frac{31350}{7}a^{10}-\frac{14176}{7}a^{9}+\frac{42650}{7}a^{8}-1431a^{7}-\frac{27611}{7}a^{6}+\frac{13473}{7}a^{5}+\frac{8839}{7}a^{4}-660a^{3}-\frac{1187}{7}a^{2}+\frac{435}{7}a+\frac{50}{7}$, $\frac{5}{7}a^{20}-\frac{50}{7}a^{19}+\frac{157}{7}a^{18}+\frac{12}{7}a^{17}-\frac{1044}{7}a^{16}+\frac{1620}{7}a^{15}+\frac{1570}{7}a^{14}-\frac{6088}{7}a^{13}+\frac{2516}{7}a^{12}+\frac{8252}{7}a^{11}-\frac{10391}{7}a^{10}-\frac{2195}{7}a^{9}+\frac{12483}{7}a^{8}-\frac{5136}{7}a^{7}-\frac{6551}{7}a^{6}+\frac{5008}{7}a^{5}+\frac{1170}{7}a^{4}-223a^{3}+\frac{93}{7}a^{2}+\frac{130}{7}a-2$, $3a^{20}-30a^{19}+\frac{668}{7}a^{18}-\frac{27}{7}a^{17}-595a^{16}+\frac{6800}{7}a^{15}+\frac{5461}{7}a^{14}-3387a^{13}+\frac{11106}{7}a^{12}+\frac{29863}{7}a^{11}-5555a^{10}-\frac{6946}{7}a^{9}+\frac{43917}{7}a^{8}-\frac{16753}{7}a^{7}-\frac{23833}{7}a^{6}+\frac{14923}{7}a^{5}+\frac{6518}{7}a^{4}-607a^{3}-\frac{838}{7}a^{2}+\frac{338}{7}a+\frac{34}{7}$, $a$, $a-1$, $2a^{20}-20a^{19}+\frac{447}{7}a^{18}-\frac{33}{7}a^{17}-391a^{16}+\frac{4556}{7}a^{15}+\frac{3376}{7}a^{14}-2220a^{13}+\frac{7911}{7}a^{12}+\frac{18860}{7}a^{11}-3717a^{10}-\frac{3056}{7}a^{9}+\frac{28341}{7}a^{8}-\frac{12310}{7}a^{7}-\frac{14388}{7}a^{6}+\frac{10351}{7}a^{5}+\frac{3299}{7}a^{4}-416a^{3}-\frac{255}{7}a^{2}+\frac{235}{7}a+\frac{12}{7}$, $\frac{2}{7}a^{20}-\frac{20}{7}a^{19}+\frac{54}{7}a^{18}+12a^{17}-\frac{629}{7}a^{16}+\frac{544}{7}a^{15}+\frac{2012}{7}a^{14}-\frac{4020}{7}a^{13}-\frac{1201}{7}a^{12}+\frac{9078}{7}a^{11}-\frac{5590}{7}a^{10}-\frac{7867}{7}a^{9}+1754a^{8}+\frac{44}{7}a^{7}-\frac{9994}{7}a^{6}+\frac{3597}{7}a^{5}+\frac{3497}{7}a^{4}-230a^{3}-\frac{433}{7}a^{2}+\frac{174}{7}a+\frac{23}{7}$, $\frac{10}{7}a^{21}-\frac{113}{7}a^{20}+\frac{450}{7}a^{19}-\frac{449}{7}a^{18}-\frac{1875}{7}a^{17}+\frac{5735}{7}a^{16}-290a^{15}-\frac{13561}{7}a^{14}+\frac{19928}{7}a^{13}+\frac{4483}{7}a^{12}-4884a^{11}+3273a^{10}+2829a^{9}-\frac{33617}{7}a^{8}+\frac{3284}{7}a^{7}+\frac{18596}{7}a^{6}-\frac{7780}{7}a^{5}-\frac{4293}{7}a^{4}+\frac{2612}{7}a^{3}+\frac{330}{7}a^{2}-35a-\frac{2}{7}$, $\frac{13}{7}a^{21}-\frac{139}{7}a^{20}+71a^{19}-\frac{246}{7}a^{18}-\frac{2707}{7}a^{17}+838a^{16}+\frac{1506}{7}a^{15}-\frac{17944}{7}a^{14}+\frac{14956}{7}a^{13}+\frac{17910}{7}a^{12}-\frac{36916}{7}a^{11}+894a^{10}+\frac{35204}{7}a^{9}-3750a^{8}-\frac{13903}{7}a^{7}+2899a^{6}+68a^{5}-\frac{6529}{7}a^{4}+\frac{1046}{7}a^{3}+\frac{876}{7}a^{2}-\frac{163}{7}a-\frac{50}{7}$, $\frac{10}{7}a^{20}-\frac{100}{7}a^{19}+\frac{317}{7}a^{18}-\frac{3}{7}a^{17}-\frac{2011}{7}a^{16}+\frac{3236}{7}a^{15}+\frac{2749}{7}a^{14}-\frac{11511}{7}a^{13}+\frac{5169}{7}a^{12}+\frac{14811}{7}a^{11}-\frac{18927}{7}a^{10}-\frac{3699}{7}a^{9}+\frac{21689}{7}a^{8}-\frac{8181}{7}a^{7}-\frac{11756}{7}a^{6}+\frac{7334}{7}a^{5}+\frac{3195}{7}a^{4}-295a^{3}-\frac{449}{7}a^{2}+\frac{192}{7}a+\frac{27}{7}$, $\frac{6}{7}a^{21}-\frac{74}{7}a^{20}+\frac{330}{7}a^{19}-\frac{439}{7}a^{18}-\frac{1254}{7}a^{17}+\frac{4901}{7}a^{16}-\frac{2866}{7}a^{15}-\frac{11630}{7}a^{14}+\frac{20596}{7}a^{13}+\frac{2585}{7}a^{12}-\frac{36549}{7}a^{11}+\frac{26974}{7}a^{10}+\frac{22716}{7}a^{9}-\frac{42648}{7}a^{8}+722a^{7}+\frac{27317}{7}a^{6}-\frac{12325}{7}a^{5}-\frac{7795}{7}a^{4}+\frac{4833}{7}a^{3}+\frac{829}{7}a^{2}-\frac{548}{7}a-\frac{38}{7}$, $\frac{18}{7}a^{21}-\frac{194}{7}a^{20}+101a^{19}-\frac{417}{7}a^{18}-\frac{3695}{7}a^{17}+1221a^{16}+\frac{1101}{7}a^{15}-\frac{25142}{7}a^{14}+\frac{23788}{7}a^{13}+3229a^{12}-\frac{54504}{7}a^{11}+\frac{14375}{7}a^{10}+\frac{48471}{7}a^{9}-6011a^{8}-\frac{15949}{7}a^{7}+\frac{30797}{7}a^{6}-\frac{1511}{7}a^{5}-\frac{9535}{7}a^{4}+256a^{3}+\frac{1079}{7}a^{2}-\frac{221}{7}a-\frac{20}{7}$, $\frac{4}{7}a^{21}-\frac{31}{7}a^{20}+\frac{29}{7}a^{19}+\frac{369}{7}a^{18}-161a^{17}-\frac{251}{7}a^{16}+\frac{5394}{7}a^{15}-\frac{5813}{7}a^{14}-1083a^{13}+\frac{19221}{7}a^{12}-\frac{4953}{7}a^{11}-3187a^{10}+\frac{23099}{7}a^{9}+\frac{6497}{7}a^{8}-\frac{22273}{7}a^{7}+\frac{5660}{7}a^{6}+\frac{8492}{7}a^{5}-\frac{4135}{7}a^{4}-151a^{3}+114a^{2}+3a-\frac{52}{7}$, $\frac{8}{7}a^{21}-\frac{81}{7}a^{20}+\frac{263}{7}a^{19}-4a^{18}-\frac{1626}{7}a^{17}+\frac{2776}{7}a^{16}+\frac{2014}{7}a^{15}-\frac{9581}{7}a^{14}+\frac{4883}{7}a^{13}+\frac{11863}{7}a^{12}-\frac{16062}{7}a^{11}-325a^{10}+\frac{17568}{7}a^{9}-\frac{7083}{7}a^{8}-\frac{9327}{7}a^{7}+\frac{5636}{7}a^{6}+\frac{2920}{7}a^{5}-\frac{1156}{7}a^{4}-\frac{725}{7}a^{3}-15a^{2}+\frac{102}{7}a+\frac{40}{7}$, $\frac{6}{7}a^{21}-\frac{53}{7}a^{20}+\frac{129}{7}a^{19}+20a^{18}-\frac{985}{7}a^{17}+\frac{584}{7}a^{16}+\frac{2578}{7}a^{15}-446a^{14}-491a^{13}+\frac{6618}{7}a^{12}+\frac{2423}{7}a^{11}-\frac{10104}{7}a^{10}+\frac{1266}{7}a^{9}+\frac{11950}{7}a^{8}-870a^{7}-\frac{8779}{7}a^{6}+\frac{6365}{7}a^{5}+\frac{3163}{7}a^{4}-\frac{2539}{7}a^{3}-\frac{426}{7}a^{2}+\frac{274}{7}a+\frac{29}{7}$, $\frac{18}{7}a^{21}-25a^{20}+\frac{517}{7}a^{19}+28a^{18}-\frac{3797}{7}a^{17}+\frac{4990}{7}a^{16}+\frac{7321}{7}a^{15}-\frac{21481}{7}a^{14}+\frac{3959}{7}a^{13}+\frac{34337}{7}a^{12}-\frac{32751}{7}a^{11}-\frac{18689}{7}a^{10}+\frac{47610}{7}a^{9}-\frac{9187}{7}a^{8}-\frac{31672}{7}a^{7}+\frac{15009}{7}a^{6}+\frac{10051}{7}a^{5}-\frac{5567}{7}a^{4}-178a^{3}+\frac{559}{7}a^{2}+\frac{31}{7}a+\frac{3}{7}$, $\frac{6}{7}a^{21}-\frac{66}{7}a^{20}+34a^{19}-\frac{86}{7}a^{18}-\frac{1459}{7}a^{17}+410a^{16}+243a^{15}-1447a^{14}+\frac{5564}{7}a^{13}+\frac{13778}{7}a^{12}-\frac{18915}{7}a^{11}-\frac{4023}{7}a^{10}+\frac{23708}{7}a^{9}-\frac{9290}{7}a^{8}-\frac{15172}{7}a^{7}+\frac{10665}{7}a^{6}+\frac{5057}{7}a^{5}-\frac{4295}{7}a^{4}-\frac{761}{7}a^{3}+\frac{625}{7}a^{2}+\frac{43}{7}a-3$, $\frac{3}{7}a^{21}-\frac{18}{7}a^{20}-\frac{26}{7}a^{19}+\frac{388}{7}a^{18}-89a^{17}-\frac{1468}{7}a^{16}+\frac{4806}{7}a^{15}-\frac{151}{7}a^{14}-\frac{12442}{7}a^{13}+\frac{10607}{7}a^{12}+\frac{12352}{7}a^{11}-\frac{23662}{7}a^{10}+215a^{9}+\frac{23776}{7}a^{8}-\frac{12648}{7}a^{7}-\frac{12584}{7}a^{6}+1351a^{5}+\frac{3484}{7}a^{4}-\frac{2553}{7}a^{3}-\frac{362}{7}a^{2}+\frac{195}{7}a-\frac{4}{7}$ (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: | \( 6449194007.94 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{1}\cdot 6449194007.94 \cdot 1}{2\cdot\sqrt{15891297445653813609959315666834999}}\cr\approx \mathstrut & 0.168529351433 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
A non-solvable group of order 81749606400 |
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed |
Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
11.11.104215884017569.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | $18{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | $18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.11.0.1}{11} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(79\) | 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
79.20.0.1 | $x^{20} - x + 70$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(1451\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(18521\) | $\Q_{18521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{18521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(71823490019\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |