Normalized defining polynomial
\( x^{20} - 3 x^{19} + x^{18} + x^{17} + 3 x^{16} + 3 x^{15} - 2 x^{14} - 22 x^{13} + 24 x^{12} - 35 x^{11} + \cdots + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(7039949498771842313261056\) \(\medspace = 2^{12}\cdot 3461^{6}\) | 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: | \(17.47\) | 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: | $2^{2/3}3461^{1/2}\approx 93.38722346966051$ | ||
Ramified primes: | \(2\), \(3461\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{211}a^{18}-\frac{95}{211}a^{17}+\frac{89}{211}a^{16}-\frac{74}{211}a^{15}-\frac{30}{211}a^{14}+\frac{94}{211}a^{13}+\frac{31}{211}a^{12}-\frac{14}{211}a^{11}+\frac{15}{211}a^{10}+\frac{76}{211}a^{9}+\frac{15}{211}a^{8}-\frac{14}{211}a^{7}+\frac{31}{211}a^{6}+\frac{94}{211}a^{5}-\frac{30}{211}a^{4}-\frac{74}{211}a^{3}+\frac{89}{211}a^{2}-\frac{95}{211}a+\frac{1}{211}$, $\frac{1}{3165}a^{19}-\frac{4}{3165}a^{18}+\frac{19}{633}a^{17}+\frac{851}{3165}a^{16}+\frac{832}{3165}a^{15}-\frac{1159}{3165}a^{14}-\frac{1543}{3165}a^{13}+\frac{162}{1055}a^{12}-\frac{279}{1055}a^{11}-\frac{458}{3165}a^{10}-\frac{1298}{3165}a^{9}-\frac{464}{1055}a^{8}-\frac{133}{1055}a^{7}-\frac{883}{3165}a^{6}+\frac{506}{3165}a^{5}-\frac{1538}{3165}a^{4}-\frac{1159}{3165}a^{3}+\frac{166}{633}a^{2}+\frac{851}{3165}a+\frac{91}{3165}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | 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: | $a$, $\frac{158}{633}a^{19}-\frac{1508}{633}a^{18}+\frac{2014}{633}a^{17}+\frac{2056}{633}a^{16}+\frac{50}{633}a^{15}-\frac{3656}{633}a^{14}-\frac{7106}{633}a^{13}-\frac{2446}{211}a^{12}+\frac{8114}{211}a^{11}-\frac{5746}{633}a^{10}+\frac{30914}{633}a^{9}-\frac{10172}{211}a^{8}-\frac{1523}{211}a^{7}-\frac{17282}{633}a^{6}+\frac{1402}{633}a^{5}+\frac{7358}{633}a^{4}+\frac{5770}{633}a^{3}+\frac{4}{633}a^{2}-\frac{74}{633}a-\frac{1690}{633}$, $\frac{1493}{3165}a^{19}-\frac{3002}{3165}a^{18}-\frac{1}{3}a^{17}-\frac{152}{3165}a^{16}+\frac{3266}{3165}a^{15}+\frac{6718}{3165}a^{14}+\frac{7411}{3165}a^{13}-\frac{7019}{1055}a^{12}+\frac{5308}{1055}a^{11}-\frac{47389}{3165}a^{10}+\frac{34886}{3165}a^{9}-\frac{9032}{1055}a^{8}+\frac{11231}{1055}a^{7}-\frac{1394}{3165}a^{6}+\frac{18673}{3165}a^{5}+\frac{1076}{3165}a^{4}+\frac{2638}{3165}a^{3}+\frac{29}{633}a^{2}+\frac{913}{3165}a-\frac{427}{3165}$, $\frac{22}{15}a^{19}-\frac{12043}{3165}a^{18}+\frac{304}{633}a^{17}+\frac{1952}{3165}a^{16}+\frac{11734}{3165}a^{15}+\frac{19892}{3165}a^{14}+\frac{5459}{3165}a^{13}-\frac{29846}{1055}a^{12}+\frac{26942}{1055}a^{11}-\frac{163976}{3165}a^{10}+\frac{211894}{3165}a^{9}-\frac{37638}{1055}a^{8}+\frac{35804}{1055}a^{7}-\frac{47971}{3165}a^{6}-\frac{9733}{3165}a^{5}+\frac{7654}{3165}a^{4}+\frac{8147}{3165}a^{3}+\frac{517}{633}a^{2}+\frac{7217}{3165}a-\frac{10988}{3165}$, $\frac{83}{211}a^{19}-\frac{236}{211}a^{18}+\frac{31}{211}a^{17}+\frac{52}{211}a^{16}+\frac{340}{211}a^{15}+\frac{515}{211}a^{14}-\frac{41}{211}a^{13}-\frac{2262}{211}a^{12}+\frac{925}{211}a^{11}-\frac{3025}{211}a^{10}+\frac{5484}{211}a^{9}-\frac{578}{211}a^{8}+\frac{2886}{211}a^{7}-\frac{2160}{211}a^{6}-\frac{2150}{211}a^{5}-\frac{1824}{211}a^{4}-\frac{122}{211}a^{3}+\frac{419}{211}a^{2}+\frac{745}{211}a+\frac{53}{211}$, $\frac{4732}{1055}a^{19}-\frac{11103}{1055}a^{18}-\frac{531}{211}a^{17}+\frac{3287}{1055}a^{16}+\frac{16789}{1055}a^{15}+\frac{24282}{1055}a^{14}+\frac{5669}{1055}a^{13}-\frac{100448}{1055}a^{12}+\frac{48486}{1055}a^{11}-\frac{129781}{1055}a^{10}+\frac{193874}{1055}a^{9}-\frac{42509}{1055}a^{8}+\frac{84952}{1055}a^{7}-\frac{55491}{1055}a^{6}-\frac{39278}{1055}a^{5}-\frac{8351}{1055}a^{4}+\frac{10197}{1055}a^{3}+\frac{2516}{211}a^{2}+\frac{9892}{1055}a-\frac{8883}{1055}$, $\frac{83}{211}a^{19}-\frac{236}{211}a^{18}+\frac{31}{211}a^{17}+\frac{52}{211}a^{16}+\frac{340}{211}a^{15}+\frac{515}{211}a^{14}-\frac{41}{211}a^{13}-\frac{2262}{211}a^{12}+\frac{925}{211}a^{11}-\frac{3025}{211}a^{10}+\frac{5484}{211}a^{9}-\frac{578}{211}a^{8}+\frac{2886}{211}a^{7}-\frac{2160}{211}a^{6}-\frac{2150}{211}a^{5}-\frac{1824}{211}a^{4}-\frac{122}{211}a^{3}+\frac{208}{211}a^{2}+\frac{745}{211}a+\frac{53}{211}$, $\frac{716}{211}a^{19}-\frac{1924}{211}a^{18}+\frac{31}{211}a^{17}+\frac{896}{211}a^{16}+\frac{2450}{211}a^{15}+\frac{3047}{211}a^{14}-\frac{674}{211}a^{13}-\frac{16610}{211}a^{12}+\frac{11475}{211}a^{11}-\frac{20116}{211}a^{10}+\frac{35446}{211}a^{9}-\frac{10284}{211}a^{8}+\frac{10693}{211}a^{7}-\frac{11022}{211}a^{6}-\frac{8058}{211}a^{5}-\frac{347}{211}a^{4}+\frac{2410}{211}a^{3}+\frac{1685}{211}a^{2}+\frac{1589}{211}a-\frac{1635}{211}$, $a^{19}-2a^{18}-a^{17}+3a^{15}+6a^{14}+4a^{13}-18a^{12}+6a^{11}-29a^{10}+30a^{9}-5a^{8}+19a^{7}-3a^{6}-5a^{5}-2a^{4}+a^{3}+2a^{2}+3a-1$, $\frac{7591}{3165}a^{19}-\frac{19819}{3165}a^{18}-\frac{2}{3}a^{17}+\frac{11336}{3165}a^{16}+\frac{28282}{3165}a^{15}+\frac{32531}{3165}a^{14}-\frac{8158}{3165}a^{13}-\frac{60223}{1055}a^{12}+\frac{35741}{1055}a^{11}-\frac{185153}{3165}a^{10}+\frac{354682}{3165}a^{9}-\frac{22814}{1055}a^{8}+\frac{28892}{1055}a^{7}-\frac{118753}{3165}a^{6}-\frac{92464}{3165}a^{5}-\frac{11783}{3165}a^{4}+\frac{30641}{3165}a^{3}+\frac{4567}{633}a^{2}+\frac{17531}{3165}a-\frac{13964}{3165}$, $\frac{789}{1055}a^{19}-\frac{1931}{1055}a^{18}-\frac{55}{211}a^{17}+\frac{819}{1055}a^{16}+\frac{2428}{1055}a^{15}+\frac{3574}{1055}a^{14}+\frac{198}{1055}a^{13}-\frac{16396}{1055}a^{12}+\frac{11372}{1055}a^{11}-\frac{21212}{1055}a^{10}+\frac{31138}{1055}a^{9}-\frac{9088}{1055}a^{8}+\frac{7749}{1055}a^{7}-\frac{4612}{1055}a^{6}-\frac{7841}{1055}a^{5}+\frac{3108}{1055}a^{4}+\frac{4534}{1055}a^{3}+\frac{15}{211}a^{2}+\frac{3299}{1055}a-\frac{2936}{1055}$ | 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: | \( 32774.4948975 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 32774.4948975 \cdot 1}{2\cdot\sqrt{7039949498771842313261056}}\cr\approx \mathstrut & 0.240038093814 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
10.2.2653290315584.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.165830644724.1, 6.2.55376.1 |
Degree 10 sibling: | 10.2.2653290315584.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 siblings: | 15.3.146928604515779584.1, deg 15 |
Degree 20 siblings: | deg 20, deg 20 |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.55376.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ | |
\(3461\) | Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $8$ | $2$ | $4$ | $4$ |