Normalized defining polynomial
\( x^{20} - 5 x^{19} + 15 x^{18} - 37 x^{17} + 78 x^{16} - 139 x^{15} + 219 x^{14} - 307 x^{13} + 386 x^{12} + \cdots + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(709000307415298179856\) \(\medspace = 2^{4}\cdot 83^{4}\cdot 983^{4}\) | 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: | \(11.03\) | 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: | $2\cdot 83^{1/2}983^{1/2}\approx 571.2757652832825$ | ||
Ramified primes: | \(2\), \(83\), \(983\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{62}a^{18}-\frac{9}{31}a^{17}-\frac{19}{62}a^{15}+\frac{15}{62}a^{14}-\frac{5}{62}a^{13}+\frac{21}{62}a^{12}-\frac{17}{62}a^{11}+\frac{14}{31}a^{10}+\frac{17}{62}a^{9}+\frac{14}{31}a^{8}-\frac{17}{62}a^{7}+\frac{21}{62}a^{6}-\frac{5}{62}a^{5}+\frac{15}{62}a^{4}-\frac{19}{62}a^{3}-\frac{9}{31}a+\frac{1}{62}$, $\frac{1}{3782}a^{19}-\frac{3}{1891}a^{18}-\frac{325}{1891}a^{17}+\frac{1345}{3782}a^{16}-\frac{1267}{3782}a^{15}-\frac{1251}{3782}a^{14}-\frac{1031}{3782}a^{13}+\frac{297}{3782}a^{12}+\frac{563}{1891}a^{11}-\frac{1507}{3782}a^{10}-\frac{845}{1891}a^{9}+\frac{1187}{3782}a^{8}-\frac{679}{3782}a^{7}+\frac{433}{3782}a^{6}+\frac{823}{3782}a^{5}-\frac{1389}{3782}a^{4}-\frac{517}{1891}a^{3}-\frac{691}{1891}a^{2}+\frac{1397}{3782}a-\frac{335}{1891}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{2634}{1891}a^{19}-\frac{12571}{1891}a^{18}+\frac{37502}{1891}a^{17}-\frac{93663}{1891}a^{16}+\frac{197977}{1891}a^{15}-\frac{353409}{1891}a^{14}+\frac{562303}{1891}a^{13}-\frac{793088}{1891}a^{12}+\frac{1001013}{1891}a^{11}-\frac{1152092}{1891}a^{10}+\frac{1206534}{1891}a^{9}-\frac{1149237}{1891}a^{8}+\frac{1002508}{1891}a^{7}-\frac{792263}{1891}a^{6}+\frac{17923}{61}a^{5}-\frac{348156}{1891}a^{4}+\frac{187669}{1891}a^{3}-\frac{81326}{1891}a^{2}+\frac{30495}{1891}a-\frac{8590}{1891}$, $\frac{599}{1891}a^{19}-\frac{2008}{1891}a^{18}+\frac{3795}{1891}a^{17}-\frac{7475}{1891}a^{16}+\frac{12717}{1891}a^{15}-\frac{14543}{1891}a^{14}+\frac{15550}{1891}a^{13}-\frac{15711}{1891}a^{12}+\frac{12136}{1891}a^{11}-\frac{13008}{1891}a^{10}+\frac{14991}{1891}a^{9}-\frac{14216}{1891}a^{8}+\frac{16375}{1891}a^{7}-\frac{21233}{1891}a^{6}+\frac{17970}{1891}a^{5}-\frac{17783}{1891}a^{4}+\frac{16132}{1891}a^{3}-\frac{9015}{1891}a^{2}+\frac{4580}{1891}a-\frac{2634}{1891}$, $\frac{5240}{1891}a^{19}-\frac{24730}{1891}a^{18}+\frac{71793}{1891}a^{17}-\frac{173929}{1891}a^{16}+\frac{358718}{1891}a^{15}-\frac{622746}{1891}a^{14}+\frac{959372}{1891}a^{13}-\frac{1307609}{1891}a^{12}+\frac{1595714}{1891}a^{11}-\frac{1774842}{1891}a^{10}+\frac{1797013}{1891}a^{9}-\frac{1653573}{1891}a^{8}+\frac{1388286}{1891}a^{7}-\frac{1050700}{1891}a^{6}+\frac{703089}{1891}a^{5}-\frac{413603}{1891}a^{4}+\frac{208663}{1891}a^{3}-\frac{84245}{1891}a^{2}+\frac{30719}{1891}a-\frac{7631}{1891}$, $\frac{1183}{3782}a^{19}-\frac{2523}{3782}a^{18}+\frac{1716}{1891}a^{17}-\frac{4869}{3782}a^{16}-\frac{564}{1891}a^{15}+\frac{12926}{1891}a^{14}-\frac{31282}{1891}a^{13}+\frac{64869}{1891}a^{12}-\frac{213133}{3782}a^{11}+\frac{289267}{3782}a^{10}-\frac{355747}{3782}a^{9}+\frac{390157}{3782}a^{8}-\frac{187122}{1891}a^{7}+\frac{168005}{1891}a^{6}-\frac{133535}{1891}a^{5}+\frac{90143}{1891}a^{4}-\frac{107471}{3782}a^{3}+\frac{27824}{1891}a^{2}-\frac{18139}{3782}a+\frac{6185}{3782}$, $\frac{3301}{3782}a^{19}-\frac{18037}{3782}a^{18}+\frac{26944}{1891}a^{17}-\frac{132593}{3782}a^{16}+\frac{140410}{1891}a^{15}-\frac{249385}{1891}a^{14}+\frac{389140}{1891}a^{13}-\frac{544514}{1891}a^{12}+\frac{1349577}{3782}a^{11}-\frac{1528839}{3782}a^{10}+\frac{1584235}{3782}a^{9}-\frac{1485825}{3782}a^{8}+\frac{638032}{1891}a^{7}-\frac{497800}{1891}a^{6}+\frac{342254}{1891}a^{5}-\frac{208632}{1891}a^{4}+\frac{225477}{3782}a^{3}-\frac{47720}{1891}a^{2}+\frac{33691}{3782}a-\frac{12559}{3782}$, $\frac{936}{1891}a^{19}-\frac{7145}{3782}a^{18}+\frac{10994}{1891}a^{17}-\frac{26960}{1891}a^{16}+\frac{107155}{3782}a^{15}-\frac{185339}{3782}a^{14}+\frac{288477}{3782}a^{13}-\frac{375547}{3782}a^{12}+\frac{453735}{3782}a^{11}-\frac{245208}{1891}a^{10}+\frac{472225}{3782}a^{9}-\frac{210289}{1891}a^{8}+\frac{334859}{3782}a^{7}-\frac{226851}{3782}a^{6}+\frac{136009}{3782}a^{5}-\frac{69257}{3782}a^{4}+\frac{17633}{3782}a^{3}-\frac{108}{1891}a^{2}+\frac{57}{1891}a+\frac{5471}{3782}$, $\frac{197}{62}a^{19}-\frac{461}{31}a^{18}+\frac{1308}{31}a^{17}-\frac{6285}{62}a^{16}+\frac{12929}{62}a^{15}-\frac{22261}{62}a^{14}+\frac{34107}{62}a^{13}-\frac{46515}{62}a^{12}+\frac{28349}{31}a^{11}-\frac{63237}{62}a^{10}+\frac{32223}{31}a^{9}-\frac{59581}{62}a^{8}+\frac{50483}{62}a^{7}-\frac{38757}{62}a^{6}+\frac{26415}{62}a^{5}-\frac{15905}{62}a^{4}+\frac{4243}{31}a^{3}-\frac{1835}{31}a^{2}+\frac{1387}{62}a-\frac{207}{31}$, $\frac{4413}{1891}a^{19}-\frac{45575}{3782}a^{18}+\frac{66138}{1891}a^{17}-\frac{159208}{1891}a^{16}+\frac{21367}{122}a^{15}-\frac{1154141}{3782}a^{14}+\frac{1766995}{3782}a^{13}-\frac{2423927}{3782}a^{12}+\frac{2955833}{3782}a^{11}-\frac{1645585}{1891}a^{10}+\frac{3351791}{3782}a^{9}-\frac{1545457}{1891}a^{8}+\frac{2595397}{3782}a^{7}-\frac{1983773}{3782}a^{6}+\frac{1334537}{3782}a^{5}-\frac{787471}{3782}a^{4}+\frac{408033}{3782}a^{3}-\frac{85386}{1891}a^{2}+\frac{28422}{1891}a-\frac{21247}{3782}$, $\frac{7059}{3782}a^{19}-\frac{13979}{1891}a^{18}+\frac{36452}{1891}a^{17}-\frac{168655}{3782}a^{16}+\frac{332273}{3782}a^{15}-\frac{17429}{122}a^{14}+\frac{800401}{3782}a^{13}-\frac{1046563}{3782}a^{12}+\frac{618241}{1891}a^{11}-\frac{1343343}{3782}a^{10}+\frac{668224}{1891}a^{9}-\frac{1198577}{3782}a^{8}+\frac{994497}{3782}a^{7}-\frac{744607}{3782}a^{6}+\frac{491943}{3782}a^{5}-\frac{292855}{3782}a^{4}+\frac{77048}{1891}a^{3}-\frac{31136}{1891}a^{2}+\frac{26271}{3782}a-\frac{1381}{1891}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 103.364702597 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 103.364702597 \cdot 1}{2\cdot\sqrt{709000307415298179856}}\cr\approx \mathstrut & 0.186130582115 \end{aligned}\]
Galois group
$C_3^5.D_6$ (as 20T669):
A non-solvable group of order 61440 |
The 126 conjugacy class representatives for $C_3^5.D_6$ |
Character table for $C_3^5.D_6$ |
Intermediate fields
5.5.81589.1, 10.2.6656764921.1, 10.4.26627059684.1, 10.4.26627059684.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 20.0.209139509314799821952907668928868954259822604297616.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(83\) | 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.6.0.1 | $x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
83.6.0.1 | $x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(983\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |