Normalized defining polynomial
\( x^{21} - 3x^{14} - 18x^{7} + 3 \)
Invariants
Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-9314970862914194811435918430261983\) \(\medspace = -\,3^{34}\cdot 7^{21}\) | 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: | \(41.46\) | 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: | \(3\), \(7\) | 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: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19}a^{14}+\frac{9}{19}a^{7}-\frac{5}{19}$, $\frac{1}{19}a^{15}+\frac{9}{19}a^{8}-\frac{5}{19}a$, $\frac{1}{19}a^{16}+\frac{9}{19}a^{9}-\frac{5}{19}a^{2}$, $\frac{1}{19}a^{17}+\frac{9}{19}a^{10}-\frac{5}{19}a^{3}$, $\frac{1}{19}a^{18}+\frac{9}{19}a^{11}-\frac{5}{19}a^{4}$, $\frac{1}{19}a^{19}+\frac{9}{19}a^{12}-\frac{5}{19}a^{5}$, $\frac{1}{19}a^{20}+\frac{9}{19}a^{13}-\frac{5}{19}a^{6}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{3}{19}a^{14}-\frac{11}{19}a^{7}-\frac{34}{19}$, $\frac{2}{19}a^{14}-\frac{1}{19}a^{7}-\frac{29}{19}$, $\frac{5}{19}a^{18}-\frac{5}{19}a^{17}+\frac{3}{19}a^{15}-\frac{3}{19}a^{14}-\frac{12}{19}a^{11}+\frac{12}{19}a^{10}-\frac{11}{19}a^{8}+\frac{11}{19}a^{7}-\frac{101}{19}a^{4}+\frac{101}{19}a^{3}-\frac{53}{19}a+\frac{34}{19}$, $\frac{5}{19}a^{20}+\frac{3}{19}a^{16}-\frac{12}{19}a^{13}-\frac{11}{19}a^{9}-\frac{101}{19}a^{6}-\frac{34}{19}a^{2}+a+1$, $\frac{5}{19}a^{20}+\frac{5}{19}a^{19}+\frac{2}{19}a^{18}-\frac{1}{19}a^{17}-\frac{1}{19}a^{16}+\frac{1}{19}a^{15}+\frac{1}{19}a^{14}-\frac{12}{19}a^{13}-\frac{12}{19}a^{12}-\frac{1}{19}a^{11}+\frac{10}{19}a^{10}+\frac{10}{19}a^{9}+\frac{9}{19}a^{8}+\frac{9}{19}a^{7}-\frac{82}{19}a^{6}-\frac{82}{19}a^{5}-\frac{29}{19}a^{4}+\frac{24}{19}a^{3}+\frac{24}{19}a^{2}-\frac{5}{19}a-\frac{5}{19}$, $\frac{16}{19}a^{20}-\frac{8}{19}a^{18}-\frac{5}{19}a^{17}-\frac{3}{19}a^{16}+\frac{3}{19}a^{14}-\frac{46}{19}a^{13}+\frac{23}{19}a^{11}+\frac{12}{19}a^{10}+\frac{11}{19}a^{9}-\frac{11}{19}a^{7}-\frac{289}{19}a^{6}+\frac{135}{19}a^{4}+\frac{101}{19}a^{3}+\frac{53}{19}a^{2}-\frac{53}{19}$, $\frac{11}{19}a^{19}+\frac{11}{19}a^{18}+\frac{5}{19}a^{17}-\frac{3}{19}a^{16}-\frac{5}{19}a^{15}-\frac{2}{19}a^{14}-\frac{34}{19}a^{12}-\frac{34}{19}a^{11}-\frac{12}{19}a^{10}+\frac{11}{19}a^{9}+\frac{12}{19}a^{8}+\frac{1}{19}a^{7}-\frac{188}{19}a^{5}-\frac{188}{19}a^{4}-\frac{101}{19}a^{3}+\frac{34}{19}a^{2}+\frac{101}{19}a+\frac{67}{19}$, $\frac{3}{19}a^{17}-\frac{2}{19}a^{16}+\frac{3}{19}a^{15}-\frac{11}{19}a^{10}+\frac{1}{19}a^{9}-\frac{11}{19}a^{8}-\frac{53}{19}a^{3}+\frac{48}{19}a^{2}-\frac{34}{19}a+1$, $\frac{16}{19}a^{20}-\frac{8}{19}a^{18}-\frac{8}{19}a^{17}-\frac{2}{19}a^{16}+\frac{3}{19}a^{15}+\frac{5}{19}a^{14}-\frac{46}{19}a^{13}+\frac{23}{19}a^{11}+\frac{23}{19}a^{10}+\frac{1}{19}a^{9}-\frac{11}{19}a^{8}-\frac{12}{19}a^{7}-\frac{289}{19}a^{6}+\frac{154}{19}a^{4}+\frac{154}{19}a^{3}+\frac{48}{19}a^{2}-\frac{53}{19}a-\frac{82}{19}$, $\frac{11}{19}a^{20}+\frac{8}{19}a^{19}+\frac{2}{19}a^{18}+\frac{3}{19}a^{17}+\frac{6}{19}a^{16}+\frac{3}{19}a^{15}-\frac{34}{19}a^{13}-\frac{23}{19}a^{12}-\frac{1}{19}a^{11}-\frac{11}{19}a^{10}-\frac{22}{19}a^{9}-\frac{11}{19}a^{8}-\frac{188}{19}a^{6}-\frac{154}{19}a^{5}-\frac{67}{19}a^{4}-\frac{53}{19}a^{3}-\frac{87}{19}a^{2}-\frac{53}{19}a-1$, $\frac{16}{19}a^{19}+\frac{11}{19}a^{18}+\frac{5}{19}a^{17}-\frac{6}{19}a^{16}-\frac{5}{19}a^{15}-\frac{2}{19}a^{14}-\frac{46}{19}a^{12}-\frac{34}{19}a^{11}-\frac{12}{19}a^{10}+\frac{22}{19}a^{9}+\frac{12}{19}a^{8}+\frac{1}{19}a^{7}-\frac{289}{19}a^{5}-\frac{188}{19}a^{4}-\frac{101}{19}a^{3}+\frac{87}{19}a^{2}+\frac{101}{19}a+\frac{67}{19}$ (assuming GRH) | 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: | \( 512291829.442 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 512291829.442 \cdot 1}{2\cdot\sqrt{9314970862914194811435918430261983}}\cr\approx \mathstrut & 0.324045119999 \end{aligned}\] (assuming GRH)
Galois group
$C_5^4:D_4$ (as 21T43):
A solvable group of order 6174 |
The 45 conjugacy class representatives for $C_5^4:D_4$ |
Character table for $C_5^4:D_4$ |
Intermediate fields
\(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 21 siblings: | data not computed |
Degree 42 siblings: | data not computed |
Minimal sibling: | 21.3.9314970862914194811435918430261983.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $21$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | $21$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $21$ | $21$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $21$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $21$ | $21$ | $1$ | $34$ | |||
\(7\) | Deg $21$ | $7$ | $3$ | $21$ |