Normalized defining polynomial
\( x^{20} + 7x^{18} + 22x^{16} + 42x^{14} + 59x^{12} + 65x^{10} + 55x^{8} + 36x^{6} + 17x^{4} + 6x^{2} + 1 \)
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: | \(908233335668669940736\) \(\medspace = 2^{10}\cdot 47^{8}\cdot 193^{2}\) | 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: | \(11.17\) | 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: | $2^{15/8}47^{1/2}193^{1/2}\approx 349.3484537310322$ | ||
Ramified primes: | \(2\), \(47\), \(193\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{22}a^{18}-\frac{1}{22}a^{16}+\frac{4}{11}a^{14}-\frac{1}{2}a^{13}-\frac{7}{22}a^{10}-\frac{1}{2}a^{5}+\frac{3}{22}a^{4}-\frac{1}{2}a^{3}+\frac{2}{11}a^{2}-\frac{1}{2}a-\frac{2}{11}$, $\frac{1}{22}a^{19}-\frac{1}{22}a^{17}-\frac{3}{22}a^{15}-\frac{1}{2}a^{14}-\frac{7}{22}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{4}{11}a^{5}-\frac{7}{22}a^{3}-\frac{1}{2}a^{2}-\frac{2}{11}a-\frac{1}{2}$
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: | $\frac{15}{11}a^{18}+\frac{95}{11}a^{16}+\frac{263}{11}a^{14}+39a^{12}+\frac{522}{11}a^{10}+45a^{8}+31a^{6}+\frac{177}{11}a^{4}+\frac{49}{11}a^{2}+\frac{17}{11}$, $\frac{5}{11}a^{18}+\frac{39}{11}a^{16}+\frac{139}{11}a^{14}+27a^{12}+\frac{438}{11}a^{10}+44a^{8}+38a^{6}+\frac{268}{11}a^{4}+\frac{119}{11}a^{2}+\frac{35}{11}$, $a^{19}+7a^{17}+22a^{15}+42a^{13}+59a^{11}+65a^{9}+55a^{7}+36a^{5}+17a^{3}+6a$, $\frac{6}{11}a^{18}+\frac{38}{11}a^{16}+\frac{103}{11}a^{14}+15a^{12}+\frac{211}{11}a^{10}+21a^{8}+17a^{6}+\frac{128}{11}a^{4}+\frac{68}{11}a^{2}+\frac{20}{11}$, $\frac{12}{11}a^{19}+\frac{5}{22}a^{18}+\frac{87}{11}a^{17}+\frac{14}{11}a^{16}+\frac{555}{22}a^{15}+\frac{31}{11}a^{14}+\frac{95}{2}a^{13}+3a^{12}+\frac{1405}{22}a^{11}+\frac{21}{11}a^{10}+\frac{135}{2}a^{9}+54a^{7}-3a^{6}+\frac{699}{22}a^{5}-\frac{95}{22}a^{4}+\frac{147}{11}a^{3}-\frac{34}{11}a^{2}+\frac{40}{11}a-\frac{31}{22}$, $\frac{16}{11}a^{19}-\frac{7}{22}a^{18}+\frac{221}{22}a^{17}-\frac{24}{11}a^{16}+\frac{663}{22}a^{15}-\frac{72}{11}a^{14}+\frac{105}{2}a^{13}-\frac{23}{2}a^{12}+\frac{1437}{22}a^{11}-\frac{325}{22}a^{10}+64a^{9}-\frac{31}{2}a^{8}+\frac{91}{2}a^{7}-\frac{25}{2}a^{6}+\frac{246}{11}a^{5}-\frac{153}{22}a^{4}+\frac{75}{11}a^{3}-\frac{61}{22}a^{2}+\frac{13}{11}a-\frac{27}{22}$, $\frac{7}{22}a^{19}-\frac{29}{22}a^{18}+\frac{24}{11}a^{17}-\frac{101}{11}a^{16}+\frac{155}{22}a^{15}-\frac{617}{22}a^{14}+\frac{29}{2}a^{13}-\frac{101}{2}a^{12}+\frac{245}{11}a^{11}-\frac{1447}{22}a^{10}+26a^{9}-\frac{135}{2}a^{8}+\frac{47}{2}a^{7}-51a^{6}+\frac{351}{22}a^{5}-\frac{302}{11}a^{4}+\frac{80}{11}a^{3}-\frac{113}{11}a^{2}+\frac{30}{11}a-\frac{49}{22}$, $\frac{39}{22}a^{19}+\frac{1}{11}a^{18}+\frac{129}{11}a^{17}+\frac{10}{11}a^{16}+\frac{376}{11}a^{15}+\frac{71}{22}a^{14}+59a^{13}+\frac{11}{2}a^{12}+\frac{1641}{22}a^{11}+\frac{107}{22}a^{10}+\frac{147}{2}a^{9}+\frac{5}{2}a^{8}+\frac{107}{2}a^{7}-a^{6}+\frac{317}{11}a^{5}-\frac{93}{22}a^{4}+\frac{111}{11}a^{3}-\frac{40}{11}a^{2}+\frac{32}{11}a-\frac{15}{11}$, $\frac{23}{22}a^{19}-\frac{43}{22}a^{18}+\frac{153}{22}a^{17}-\frac{287}{22}a^{16}+\frac{224}{11}a^{15}-\frac{839}{22}a^{14}+\frac{71}{2}a^{13}-66a^{12}+\frac{1005}{22}a^{11}-\frac{1855}{22}a^{10}+46a^{9}-85a^{8}+34a^{7}-\frac{125}{2}a^{6}+\frac{205}{11}a^{5}-\frac{378}{11}a^{4}+\frac{125}{22}a^{3}-\frac{271}{22}a^{2}+\frac{29}{22}a-\frac{35}{11}$ | 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: | \( 99.8204300191 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 99.8204300191 \cdot 1}{2\cdot\sqrt{908233335668669940736}}\cr\approx \mathstrut & 0.158814227074 \end{aligned}\]
Galois group
$C_2^6.C_2^8:D_5$ (as 20T853):
A solvable group of order 163840 |
The 280 conjugacy class representatives for $C_2^6.C_2^8:D_5$ |
Character table for $C_2^6.C_2^8:D_5$ |
Intermediate fields
5.1.2209.1, 10.2.941778433.1 |
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: | 20.0.4818813138470041550848.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
2.10.10.5 | $x^{10} + 34 x^{8} - 24 x^{7} + 368 x^{6} - 496 x^{5} + 1568 x^{4} - 1760 x^{3} + 2992 x^{2} - 1856 x + 352$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ | |
\(47\) | 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.8.4.1 | $x^{8} + 204 x^{6} + 80 x^{5} + 14080 x^{4} - 6880 x^{3} + 384824 x^{2} - 499680 x + 3453444$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(193\) | 193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
193.4.2.1 | $x^{4} + 62916 x^{3} + 995609232 x^{2} + 188857096344 x + 6083572132$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
193.4.0.1 | $x^{4} + 6 x^{2} + 148 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
193.4.0.1 | $x^{4} + 6 x^{2} + 148 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |