Normalized defining polynomial
\( x^{21} - 9x^{14} - 12x^{7} + 1 \)
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: | \(-108608979330127274981177223\) \(\medspace = -\,3^{28}\cdot 7^{15}\) | 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: | \(17.37\) | 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: | $3^{4/3}7^{5/6}\approx 21.898281770364438$ | ||
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: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17}a^{14}+\frac{3}{17}a^{7}+\frac{7}{17}$, $\frac{1}{17}a^{15}+\frac{3}{17}a^{8}+\frac{7}{17}a$, $\frac{1}{17}a^{16}+\frac{3}{17}a^{9}+\frac{7}{17}a^{2}$, $\frac{1}{17}a^{17}+\frac{3}{17}a^{10}+\frac{7}{17}a^{3}$, $\frac{1}{119}a^{18}+\frac{2}{119}a^{17}+\frac{2}{119}a^{16}-\frac{1}{119}a^{15}-\frac{1}{119}a^{14}-\frac{3}{7}a^{13}+\frac{2}{7}a^{12}-\frac{48}{119}a^{11}+\frac{6}{119}a^{10}-\frac{45}{119}a^{9}-\frac{54}{119}a^{8}-\frac{3}{119}a^{7}-\frac{1}{7}a^{5}+\frac{41}{119}a^{4}-\frac{3}{119}a^{3}+\frac{48}{119}a^{2}-\frac{58}{119}a-\frac{41}{119}$, $\frac{1}{119}a^{19}-\frac{2}{119}a^{17}+\frac{2}{119}a^{16}+\frac{1}{119}a^{15}+\frac{1}{7}a^{13}+\frac{3}{119}a^{12}-\frac{1}{7}a^{11}-\frac{57}{119}a^{10}+\frac{57}{119}a^{9}-\frac{2}{17}a^{8}+\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{44}{119}a^{5}+\frac{2}{7}a^{4}+\frac{54}{119}a^{3}+\frac{2}{17}a^{2}-\frac{44}{119}a-\frac{3}{7}$, $\frac{1}{119}a^{20}-\frac{1}{119}a^{17}-\frac{2}{119}a^{16}-\frac{2}{119}a^{15}+\frac{1}{119}a^{14}+\frac{20}{119}a^{13}+\frac{3}{7}a^{12}-\frac{2}{7}a^{11}+\frac{48}{119}a^{10}-\frac{6}{119}a^{9}+\frac{45}{119}a^{8}+\frac{54}{119}a^{7}-\frac{44}{119}a^{6}+\frac{1}{7}a^{4}-\frac{41}{119}a^{3}+\frac{3}{119}a^{2}-\frac{48}{119}a+\frac{58}{119}$
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: | $\frac{7}{17}a^{18}-\frac{64}{17}a^{11}-\frac{70}{17}a^{4}$, $\frac{7}{17}a^{19}-\frac{64}{17}a^{12}-\frac{70}{17}a^{5}$, $\frac{41}{119}a^{20}+\frac{33}{119}a^{19}+\frac{11}{119}a^{18}+\frac{6}{119}a^{17}+\frac{6}{119}a^{16}-\frac{11}{119}a^{15}+\frac{9}{119}a^{14}-\frac{370}{119}a^{13}-\frac{292}{119}a^{12}-\frac{103}{119}a^{11}-\frac{50}{119}a^{10}-\frac{50}{119}a^{9}+\frac{103}{119}a^{8}-\frac{92}{119}a^{7}-\frac{461}{119}a^{6}-\frac{449}{119}a^{5}-\frac{110}{119}a^{4}-\frac{128}{119}a^{3}-\frac{11}{17}a^{2}+\frac{93}{119}a-\frac{22}{119}$, $\frac{33}{119}a^{20}+\frac{11}{119}a^{19}+\frac{6}{119}a^{18}-\frac{15}{119}a^{17}-\frac{11}{119}a^{16}-\frac{19}{119}a^{15}+\frac{6}{119}a^{14}-\frac{292}{119}a^{13}-\frac{103}{119}a^{12}-\frac{50}{119}a^{11}+\frac{125}{119}a^{10}+\frac{103}{119}a^{9}+\frac{181}{119}a^{8}-\frac{67}{119}a^{7}-\frac{449}{119}a^{6}-\frac{110}{119}a^{5}-\frac{128}{119}a^{4}+\frac{36}{17}a^{3}+\frac{93}{119}a^{2}+\frac{139}{119}a+\frac{8}{119}$, $\frac{40}{119}a^{20}-\frac{46}{119}a^{19}-\frac{30}{119}a^{18}-\frac{8}{119}a^{17}-\frac{1}{119}a^{16}-\frac{5}{119}a^{15}-\frac{356}{119}a^{13}+\frac{58}{17}a^{12}+\frac{267}{119}a^{11}+\frac{78}{119}a^{10}+\frac{2}{17}a^{9}+\frac{53}{119}a^{8}-\frac{502}{119}a^{6}+\frac{90}{17}a^{5}+\frac{55}{17}a^{4}+\frac{12}{119}a^{3}+\frac{10}{119}a^{2}-\frac{18}{119}a-\frac{4}{7}$, $\frac{55}{119}a^{20}+\frac{3}{17}a^{19}+\frac{1}{119}a^{18}+\frac{24}{119}a^{17}+\frac{4}{119}a^{16}+\frac{8}{119}a^{15}+\frac{5}{119}a^{14}-\frac{498}{119}a^{13}-\frac{192}{119}a^{12}-\frac{2}{17}a^{11}-\frac{31}{17}a^{10}-\frac{39}{119}a^{9}-\frac{78}{119}a^{8}-\frac{36}{119}a^{7}-\frac{635}{119}a^{6}-\frac{227}{119}a^{5}+\frac{24}{119}a^{4}-\frac{291}{119}a^{3}+\frac{45}{119}a^{2}-\frac{80}{119}a-\frac{50}{119}$, $\frac{26}{119}a^{20}-\frac{53}{119}a^{19}-\frac{24}{119}a^{18}+\frac{11}{119}a^{17}+\frac{4}{119}a^{16}+\frac{10}{119}a^{15}+\frac{1}{119}a^{14}-\frac{228}{119}a^{13}+\frac{470}{119}a^{12}+\frac{31}{17}a^{11}-\frac{103}{119}a^{10}-\frac{39}{119}a^{9}-\frac{89}{119}a^{8}+\frac{3}{119}a^{7}-\frac{362}{119}a^{6}+\frac{717}{119}a^{5}+\frac{274}{119}a^{4}-\frac{76}{119}a^{3}+\frac{11}{119}a^{2}-\frac{100}{119}a-\frac{27}{119}$, $\frac{23}{119}a^{20}+\frac{29}{119}a^{19}+\frac{4}{119}a^{18}+\frac{25}{119}a^{17}+\frac{13}{119}a^{16}-\frac{2}{119}a^{14}-\frac{29}{17}a^{13}-\frac{270}{119}a^{12}-\frac{39}{119}a^{11}-\frac{33}{17}a^{10}-\frac{114}{119}a^{9}+\frac{11}{119}a^{7}-\frac{45}{17}a^{6}-\frac{39}{17}a^{5}-\frac{6}{119}a^{4}-\frac{250}{119}a^{3}-\frac{215}{119}a^{2}+\frac{2}{7}a+\frac{20}{119}$, $\frac{2}{7}a^{20}+\frac{20}{119}a^{19}-\frac{23}{119}a^{18}-\frac{29}{119}a^{17}-\frac{4}{119}a^{16}+\frac{3}{119}a^{15}+\frac{8}{119}a^{14}-\frac{18}{7}a^{13}-\frac{178}{119}a^{12}+\frac{29}{17}a^{11}+\frac{270}{119}a^{10}+\frac{39}{119}a^{9}-\frac{6}{17}a^{8}-\frac{61}{119}a^{7}-\frac{24}{7}a^{6}-\frac{251}{119}a^{5}+\frac{45}{17}a^{4}+\frac{39}{17}a^{3}+\frac{6}{119}a^{2}+\frac{89}{119}a-\frac{114}{119}$, $\frac{79}{119}a^{20}+\frac{41}{119}a^{19}+\frac{2}{17}a^{18}+\frac{1}{17}a^{17}-\frac{6}{119}a^{16}+\frac{9}{119}a^{15}-\frac{5}{119}a^{14}-\frac{698}{119}a^{13}-\frac{370}{119}a^{12}-\frac{128}{119}a^{11}-\frac{64}{119}a^{10}+\frac{50}{119}a^{9}-\frac{92}{119}a^{8}+\frac{53}{119}a^{7}-\frac{1079}{119}a^{6}-\frac{495}{119}a^{5}-\frac{20}{17}a^{4}-\frac{87}{119}a^{3}+\frac{111}{119}a^{2}-\frac{73}{119}a-\frac{1}{119}$, $\frac{19}{119}a^{20}+\frac{4}{119}a^{19}+\frac{12}{119}a^{18}+\frac{25}{119}a^{17}-\frac{6}{119}a^{16}-\frac{4}{119}a^{15}+\frac{1}{17}a^{14}-\frac{164}{119}a^{13}-\frac{39}{119}a^{12}-\frac{100}{119}a^{11}-\frac{33}{17}a^{10}+\frac{50}{119}a^{9}+\frac{39}{119}a^{8}-\frac{64}{119}a^{7}-\frac{309}{119}a^{6}-\frac{23}{119}a^{5}-\frac{239}{119}a^{4}-\frac{284}{119}a^{3}+\frac{94}{119}a^{2}-\frac{62}{119}a-\frac{10}{17}$ (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: | \( 51151.7616302 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 51151.7616302 \cdot 1}{2\cdot\sqrt{108608979330127274981177223}}\cr\approx \mathstrut & 0.299644655153 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times F_7$ (as 21T9):
A solvable group of order 126 |
The 21 conjugacy class representatives for $C_3\times F_7$ |
Character table for $C_3\times F_7$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 7.1.110270727.1 |
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: | 21.3.65701728236743660173798567.1 |
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 | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.3.0.1}{3} }^{7}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.3.0.1}{3} }^{7}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
Deg $18$ | $3$ | $6$ | $24$ | ||||
\(7\) | 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
7.18.15.5 | $x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |