Normalized defining polynomial
\( x^{21} - 5x^{14} - 8x^{7} - 1 \)
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: | \(-3219905755813179726837607\) \(\medspace = -\,7^{29}\) | 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: | \(14.69\) | 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: | $7^{59/42}\approx 15.387268511758752$ | ||
Ramified primes: | \(7\) | 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: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7}a^{11}-\frac{2}{7}a^{10}-\frac{2}{7}a^{9}+\frac{3}{7}a^{8}-\frac{3}{7}a^{7}+\frac{3}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}+\frac{3}{7}a^{5}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{3}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{14}-\frac{1}{7}a^{7}+\frac{2}{7}$, $\frac{1}{7}a^{15}-\frac{1}{7}a^{8}+\frac{2}{7}a$, $\frac{1}{7}a^{16}-\frac{1}{7}a^{9}+\frac{2}{7}a^{2}$, $\frac{1}{7}a^{17}-\frac{1}{7}a^{10}+\frac{2}{7}a^{3}$, $\frac{1}{7}a^{18}-\frac{2}{7}a^{10}-\frac{2}{7}a^{9}+\frac{3}{7}a^{8}-\frac{3}{7}a^{7}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{19}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{5}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{20}+\frac{1}{7}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{3}{7}a^{7}-\frac{2}{7}a^{6}+\frac{3}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{2}{7}$
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: | $\frac{3}{7}a^{16}-\frac{17}{7}a^{9}-\frac{15}{7}a^{2}$, $\frac{3}{7}a^{17}-\frac{17}{7}a^{10}-\frac{15}{7}a^{3}$, $\frac{3}{7}a^{19}+\frac{1}{7}a^{18}+\frac{2}{7}a^{16}+\frac{1}{7}a^{14}-\frac{16}{7}a^{12}-\frac{4}{7}a^{11}-\frac{11}{7}a^{9}+\frac{1}{7}a^{8}-\frac{5}{7}a^{7}-\frac{19}{7}a^{5}-2a^{4}-\frac{9}{7}a^{2}-\frac{4}{7}a-\frac{3}{7}$, $\frac{3}{7}a^{20}-\frac{3}{7}a^{18}+\frac{1}{7}a^{17}+\frac{1}{7}a^{15}-\frac{16}{7}a^{13}+\frac{16}{7}a^{11}-\frac{5}{7}a^{10}-\frac{6}{7}a^{8}-\frac{1}{7}a^{7}-\frac{19}{7}a^{6}+\frac{19}{7}a^{4}-\frac{10}{7}a^{3}-\frac{6}{7}a+\frac{4}{7}$, $\frac{3}{7}a^{20}-\frac{3}{7}a^{19}+\frac{1}{7}a^{15}+\frac{1}{7}a^{14}-\frac{16}{7}a^{13}+\frac{16}{7}a^{12}-\frac{1}{7}a^{9}-\frac{6}{7}a^{8}-\frac{6}{7}a^{7}-\frac{19}{7}a^{6}+\frac{19}{7}a^{5}+\frac{4}{7}a^{2}-\frac{6}{7}a-\frac{6}{7}$, $\frac{3}{7}a^{19}-\frac{1}{7}a^{18}+\frac{2}{7}a^{16}-\frac{2}{7}a^{15}+\frac{1}{7}a^{14}-\frac{16}{7}a^{12}+\frac{5}{7}a^{11}-\frac{11}{7}a^{9}+\frac{10}{7}a^{8}-\frac{5}{7}a^{7}-\frac{19}{7}a^{5}+\frac{10}{7}a^{4}-\frac{9}{7}a^{2}+\frac{13}{7}a-\frac{3}{7}$, $\frac{3}{7}a^{20}-\frac{3}{7}a^{19}-\frac{3}{7}a^{17}-\frac{16}{7}a^{13}+\frac{15}{7}a^{12}+\frac{16}{7}a^{10}-\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{19}{7}a^{6}+\frac{23}{7}a^{5}+\frac{19}{7}a^{3}+\frac{4}{7}a-\frac{4}{7}$, $\frac{3}{7}a^{20}+\frac{1}{7}a^{17}-\frac{1}{7}a^{16}-\frac{1}{7}a^{14}-\frac{16}{7}a^{13}+\frac{1}{7}a^{12}-\frac{6}{7}a^{10}+\frac{5}{7}a^{9}+\frac{1}{7}a^{8}+\frac{5}{7}a^{7}-\frac{19}{7}a^{6}-\frac{4}{7}a^{5}-\frac{6}{7}a^{3}+\frac{10}{7}a^{2}-\frac{4}{7}a+\frac{3}{7}$, $\frac{4}{7}a^{20}+\frac{2}{7}a^{19}+\frac{1}{7}a^{18}-\frac{1}{7}a^{17}-\frac{1}{7}a^{14}-\frac{20}{7}a^{13}-\frac{11}{7}a^{12}-\frac{5}{7}a^{11}+\frac{5}{7}a^{10}+\frac{5}{7}a^{7}-\frac{33}{7}a^{6}-\frac{9}{7}a^{5}-\frac{10}{7}a^{4}+\frac{10}{7}a^{3}+\frac{3}{7}$, $\frac{3}{7}a^{20}+\frac{3}{7}a^{19}+\frac{1}{7}a^{17}+\frac{1}{7}a^{16}-\frac{1}{7}a^{15}-\frac{1}{7}a^{14}-\frac{16}{7}a^{13}-\frac{15}{7}a^{12}-\frac{5}{7}a^{10}-\frac{5}{7}a^{9}+\frac{5}{7}a^{8}+\frac{6}{7}a^{7}-\frac{19}{7}a^{6}-\frac{23}{7}a^{5}-\frac{10}{7}a^{3}-\frac{3}{7}a^{2}+\frac{3}{7}a+\frac{6}{7}$, $\frac{1}{7}a^{19}-\frac{1}{7}a^{18}+\frac{1}{7}a^{17}+\frac{1}{7}a^{16}-\frac{1}{7}a^{14}-\frac{5}{7}a^{12}+\frac{5}{7}a^{11}-\frac{6}{7}a^{10}-\frac{5}{7}a^{9}+\frac{6}{7}a^{7}-\frac{10}{7}a^{5}+\frac{10}{7}a^{4}-\frac{6}{7}a^{3}-\frac{3}{7}a^{2}+\frac{6}{7}$ (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: | \( 6020.25035863 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 6020.25035863 \cdot 1}{2\cdot\sqrt{3219905755813179726837607}}\cr\approx \mathstrut & 0.204819724945 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times D_7$ (as 21T3):
A solvable group of order 42 |
The 15 conjugacy class representatives for $C_3\times D_7$ |
Character table for $C_3\times D_7$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 7.1.40353607.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 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 | $21$ | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | $21$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $21$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $21$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $21$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | Deg $21$ | $21$ | $1$ | $29$ |