Normalized defining polynomial
\( x^{20} + 3x^{18} + 4x^{16} + x^{14} - 3x^{12} - 6x^{10} - 9x^{8} - 13x^{6} - 12x^{4} - 6x^{2} - 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-2948433986992424882176\) \(\medspace = -\,2^{12}\cdot 41^{2}\cdot 4549^{4}\) | 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: | \(11.84\) | 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: | $2^{63/32}41^{1/2}4549^{1/2}\approx 1690.4514439867726$ | ||
Ramified primes: | \(2\), \(41\), \(4549\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $10$ | 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: | $a$, $a^{18}+2a^{16}+a^{14}-2a^{12}-2a^{10}-2a^{8}-5a^{6}-6a^{4}-2a^{2}+1$, $a^{18}-2a^{14}-3a^{12}+a^{10}+3a^{4}+7a^{2}+3$, $2a^{18}+4a^{16}+4a^{14}-a^{12}-4a^{10}-8a^{8}-12a^{6}-15a^{4}-11a^{2}-4$, $a^{18}+2a^{16}+2a^{14}-a^{12}-2a^{10}-4a^{8}-5a^{6}-8a^{4}-5a^{2}-2$, $a^{19}+\frac{5}{2}a^{18}+4a^{17}+\frac{13}{2}a^{16}+\frac{9}{2}a^{15}+\frac{13}{2}a^{14}+\frac{1}{2}a^{13}-\frac{3}{2}a^{12}-5a^{11}-\frac{15}{2}a^{10}-6a^{9}-\frac{21}{2}a^{8}-\frac{21}{2}a^{7}-17a^{6}-\frac{29}{2}a^{5}-\frac{47}{2}a^{4}-\frac{25}{2}a^{3}-\frac{33}{2}a^{2}-4a-4$, $\frac{5}{2}a^{19}+\frac{13}{2}a^{17}-\frac{1}{2}a^{16}+\frac{13}{2}a^{15}-\frac{3}{2}a^{14}-\frac{3}{2}a^{13}-a^{12}-\frac{15}{2}a^{11}+a^{10}-\frac{21}{2}a^{9}+\frac{3}{2}a^{8}-17a^{7}+\frac{3}{2}a^{6}-\frac{47}{2}a^{5}+\frac{7}{2}a^{4}-\frac{33}{2}a^{3}+5a^{2}-4a+2$, $3a^{19}+\frac{1}{2}a^{18}+\frac{13}{2}a^{17}+\frac{3}{2}a^{16}+6a^{15}+a^{14}-2a^{13}-a^{12}-\frac{13}{2}a^{11}-\frac{3}{2}a^{10}-\frac{23}{2}a^{9}-\frac{3}{2}a^{8}-18a^{7}-\frac{7}{2}a^{6}-\frac{47}{2}a^{5}-5a^{4}-16a^{3}-2a^{2}-\frac{11}{2}a$, $a^{18}+a^{17}+\frac{5}{2}a^{16}+2a^{15}+\frac{5}{2}a^{14}+\frac{3}{2}a^{13}-\frac{1}{2}a^{12}-\frac{3}{2}a^{11}-\frac{5}{2}a^{10}-2a^{9}-\frac{9}{2}a^{8}-3a^{7}-\frac{13}{2}a^{6}-\frac{11}{2}a^{5}-9a^{4}-\frac{13}{2}a^{3}-\frac{13}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}$, $2a^{19}-a^{18}+5a^{17}-\frac{3}{2}a^{16}+5a^{15}-\frac{3}{2}a^{14}-\frac{3}{2}a^{13}+\frac{1}{2}a^{12}-\frac{11}{2}a^{11}+\frac{3}{2}a^{10}-8a^{9}+\frac{7}{2}a^{8}-13a^{7}+\frac{9}{2}a^{6}-\frac{37}{2}a^{5}+6a^{4}-\frac{25}{2}a^{3}+\frac{9}{2}a^{2}-\frac{7}{2}a+\frac{3}{2}$ | 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: | \( 265.437747416 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 265.437747416 \cdot 1}{2\cdot\sqrt{2948433986992424882176}}\cr\approx \mathstrut & 0.149216136022 \end{aligned}\]
Galois group
$C_2^{10}.C_2\wr S_5$ (as 20T1015):
A non-solvable group of order 3932160 |
The 506 conjugacy class representatives for $C_2^{10}.C_2\wr S_5$ |
Character table for $C_2^{10}.C_2\wr S_5$ |
Intermediate fields
5.1.4549.1, 10.2.848429441.1 |
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: | 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 | R | $20$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
2.12.12.15 | $x^{12} - 4 x^{11} + 44 x^{10} - 152 x^{9} + 692 x^{8} - 1952 x^{7} + 6624 x^{6} - 12224 x^{5} + 33776 x^{4} - 54976 x^{3} + 73920 x^{2} - 129664 x + 96448$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
41.12.0.1 | $x^{12} + 26 x^{6} + 13 x^{5} + 34 x^{4} + 24 x^{3} + 21 x^{2} + 27 x + 6$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(4549\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |