Normalized defining polynomial
\( x^{20} + 2x^{18} + x^{14} + 7x^{12} - 6x^{10} + 7x^{8} + x^{6} + 2x^{2} + 1 \)
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: | \(960210593667613720576\) \(\medspace = 2^{20}\cdot 5501^{4}\) | 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: | \(11.20\) | 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: | not computed | ||
Ramified primes: | \(2\), \(5501\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{13}-\frac{1}{8}a^{12}+\frac{3}{8}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{3}{8}a^{6}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a+\frac{1}{8}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{14}-\frac{1}{8}a^{13}+\frac{1}{4}a^{12}+\frac{3}{8}a^{11}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{3}{8}a^{7}+\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{1}{4}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{16}a^{18}-\frac{1}{16}a^{16}-\frac{1}{4}a^{15}+\frac{3}{16}a^{14}-\frac{1}{4}a^{13}-\frac{1}{16}a^{10}+\frac{1}{4}a^{9}-\frac{3}{16}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{16}a^{4}+\frac{1}{4}a^{3}-\frac{3}{16}a^{2}+\frac{1}{4}a+\frac{3}{16}$, $\frac{1}{16}a^{19}-\frac{1}{16}a^{17}+\frac{3}{16}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{16}a^{11}-\frac{3}{16}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}+\frac{1}{16}a^{5}-\frac{3}{16}a^{3}+\frac{1}{4}a^{2}+\frac{3}{16}a+\frac{1}{4}$
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: | $a^{19}+2a^{17}-\frac{3}{2}a^{15}-\frac{5}{2}a^{13}+7a^{11}-\frac{9}{2}a^{9}-\frac{5}{2}a^{7}+5a^{5}-\frac{5}{2}a^{3}-\frac{3}{2}a$, $\frac{5}{8}a^{19}+\frac{13}{8}a^{17}+\frac{3}{8}a^{15}-\frac{3}{4}a^{13}+\frac{25}{8}a^{11}-\frac{15}{8}a^{9}+\frac{1}{4}a^{7}+\frac{19}{8}a^{5}-\frac{3}{8}a^{3}+\frac{5}{8}a$, $a$, $\frac{1}{8}a^{19}+\frac{9}{8}a^{17}+\frac{19}{8}a^{15}+\frac{3}{4}a^{13}-\frac{3}{8}a^{11}+\frac{33}{8}a^{9}-\frac{1}{4}a^{7}-\frac{9}{8}a^{5}+\frac{29}{8}a^{3}+\frac{5}{8}a$, $\frac{1}{16}a^{19}-\frac{7}{8}a^{18}+\frac{3}{16}a^{17}-\frac{5}{4}a^{16}-\frac{9}{16}a^{15}+\frac{13}{8}a^{14}-\frac{3}{2}a^{13}+\frac{3}{8}a^{12}+\frac{11}{16}a^{11}-\frac{13}{2}a^{10}+\frac{17}{16}a^{9}+\frac{57}{8}a^{8}-4a^{7}-\frac{41}{8}a^{6}+\frac{45}{16}a^{5}+\frac{1}{4}a^{4}-\frac{7}{16}a^{3}+\frac{3}{8}a^{2}-\frac{21}{16}a$, $\frac{1}{8}a^{19}+\frac{23}{16}a^{18}+\frac{7}{4}a^{17}+\frac{67}{16}a^{16}+\frac{31}{8}a^{15}+\frac{33}{16}a^{14}+\frac{9}{8}a^{13}-\frac{11}{8}a^{12}-\frac{1}{2}a^{11}+\frac{119}{16}a^{10}+\frac{63}{8}a^{9}+\frac{3}{16}a^{8}-\frac{11}{8}a^{7}+\frac{5}{8}a^{6}+\frac{9}{4}a^{5}+\frac{61}{16}a^{4}+\frac{25}{8}a^{3}+\frac{31}{16}a^{2}+\frac{9}{4}a+\frac{11}{16}$, $\frac{1}{8}a^{19}+\frac{1}{2}a^{18}+\frac{7}{8}a^{17}+\frac{7}{8}a^{16}+\frac{21}{8}a^{15}-\frac{1}{2}a^{14}+\frac{11}{4}a^{13}-\frac{3}{8}a^{12}-\frac{1}{8}a^{11}+\frac{21}{8}a^{10}+\frac{11}{8}a^{9}-\frac{15}{4}a^{8}+\frac{23}{4}a^{7}+\frac{25}{8}a^{6}-\frac{23}{8}a^{5}-\frac{7}{8}a^{4}+\frac{27}{8}a^{3}+\frac{1}{2}a^{2}+\frac{21}{8}a-\frac{1}{8}$, $\frac{1}{16}a^{19}+\frac{1}{16}a^{18}+\frac{3}{16}a^{17}+\frac{13}{16}a^{16}+\frac{3}{16}a^{15}+\frac{31}{16}a^{14}+\frac{1}{4}a^{13}+\frac{7}{8}a^{12}+\frac{11}{16}a^{11}-\frac{7}{16}a^{10}+\frac{5}{16}a^{9}+\frac{45}{16}a^{8}+\frac{3}{4}a^{7}-\frac{1}{8}a^{6}+\frac{13}{16}a^{5}+\frac{3}{16}a^{4}+\frac{13}{16}a^{3}+\frac{17}{16}a^{2}+\frac{15}{16}a+\frac{21}{16}$, $\frac{1}{16}a^{19}+\frac{3}{2}a^{18}+\frac{3}{16}a^{17}+\frac{29}{8}a^{16}-\frac{9}{16}a^{15}-\frac{1}{4}a^{14}-\frac{3}{2}a^{13}-\frac{23}{8}a^{12}+\frac{11}{16}a^{11}+\frac{71}{8}a^{10}+\frac{17}{16}a^{9}-\frac{7}{2}a^{8}-4a^{7}-\frac{11}{8}a^{6}+\frac{45}{16}a^{5}+\frac{43}{8}a^{4}-\frac{7}{16}a^{3}-\frac{3}{4}a^{2}-\frac{21}{16}a-\frac{13}{8}$ | 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: | \( 103.153871476 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 103.153871476 \cdot 1}{2\cdot\sqrt{960210593667613720576}}\cr\approx \mathstrut & 0.159613997127 \end{aligned}\]
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
5.1.5501.1, 10.2.1936704064.1, 10.0.30987265024.2, 10.0.484176016.1 |
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: | 20.0.26608253089927391133564988913207271752728576.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
2.12.12.11 | $x^{12} + 28 x^{10} + 40 x^{9} + 356 x^{8} + 896 x^{7} + 2720 x^{6} + 6656 x^{5} + 12464 x^{4} + 19456 x^{3} + 26304 x^{2} + 19840 x + 5824$ | $2$ | $6$ | $12$ | $A_4 \times C_2$ | $[2, 2]^{6}$ | |
\(5501\) | 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 $4$ | $2$ | $2$ | $2$ |