Normalized defining polynomial
\( x^{12} - 6 x^{11} + 12 x^{10} - 3 x^{9} - 18 x^{8} - 3 x^{7} + 67 x^{6} - 39 x^{5} - 39 x^{4} - 12 x^{3} + \cdots + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(32357746661769\) \(\medspace = 3^{18}\cdot 17^{4}\) | 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: | \(13.36\) | 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: | $3^{3/2}17^{1/2}\approx 21.42428528562855$ | ||
Ramified primes: | \(3\), \(17\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{111}a^{9}-\frac{9}{37}a^{8}+\frac{14}{37}a^{7}+\frac{46}{111}a^{6}+\frac{10}{37}a^{5}-\frac{3}{37}a^{4}+\frac{44}{111}a^{3}-\frac{6}{37}a^{2}+\frac{11}{37}a+\frac{47}{111}$, $\frac{1}{111}a^{10}-\frac{7}{37}a^{8}-\frac{41}{111}a^{7}+\frac{17}{37}a^{6}+\frac{8}{37}a^{5}+\frac{23}{111}a^{4}-\frac{17}{37}a^{3}-\frac{3}{37}a^{2}+\frac{50}{111}a+\frac{16}{37}$, $\frac{1}{333}a^{11}+\frac{1}{333}a^{10}+\frac{1}{333}a^{9}+\frac{10}{333}a^{8}-\frac{65}{333}a^{7}-\frac{134}{333}a^{6}+\frac{152}{333}a^{5}+\frac{107}{333}a^{4}-\frac{91}{333}a^{3}-\frac{133}{333}a^{2}+\frac{158}{333}a+\frac{83}{333}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{113}{333} a^{11} + \frac{679}{333} a^{10} - \frac{1409}{333} a^{9} + \frac{580}{333} a^{8} + \frac{1693}{333} a^{7} + \frac{247}{333} a^{6} - \frac{7078}{333} a^{5} + \frac{5135}{333} a^{4} + \frac{2444}{333} a^{3} + \frac{1259}{333} a^{2} - \frac{3706}{333} a - \frac{307}{333} \) (order $18$) | 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: | $\frac{286}{333}a^{11}-\frac{1772}{333}a^{10}+\frac{3763}{333}a^{9}-\frac{1514}{333}a^{8}-\frac{4961}{333}a^{7}+\frac{10}{333}a^{6}+\frac{19469}{333}a^{5}-\frac{14725}{333}a^{4}-\frac{9172}{333}a^{3}-\frac{1849}{333}a^{2}+\frac{11408}{333}a+\frac{1460}{333}$, $\frac{1}{333}a^{11}-\frac{53}{333}a^{10}+\frac{241}{333}a^{9}-\frac{341}{333}a^{8}-\frac{92}{333}a^{7}+\frac{493}{333}a^{6}+\frac{728}{333}a^{5}-\frac{1963}{333}a^{4}-\frac{97}{333}a^{3}+\frac{29}{333}a^{2}+\frac{1382}{333}a+\frac{113}{333}$, $\frac{76}{333}a^{11}-\frac{422}{333}a^{10}+\frac{766}{333}a^{9}-\frac{86}{333}a^{8}-\frac{1163}{333}a^{7}-\frac{512}{333}a^{6}+\frac{4316}{333}a^{5}-\frac{1873}{333}a^{4}-\frac{1774}{333}a^{3}-\frac{1063}{333}a^{2}+\frac{1220}{333}a+\frac{182}{333}$, $\frac{238}{111}a^{11}-\frac{1462}{111}a^{10}+\frac{3064}{111}a^{9}-\frac{1148}{111}a^{8}-\frac{4123}{111}a^{7}-\frac{140}{111}a^{6}+\frac{15998}{111}a^{5}-\frac{11540}{111}a^{4}-\frac{7750}{111}a^{3}-\frac{1732}{111}a^{2}+\frac{8899}{111}a+\frac{1601}{111}$, $\frac{14}{333}a^{11}-\frac{82}{333}a^{10}+\frac{155}{333}a^{9}+\frac{14}{333}a^{8}-\frac{376}{333}a^{7}+\frac{47}{333}a^{6}+\frac{1057}{333}a^{5}-\frac{647}{333}a^{4}-\frac{1163}{333}a^{3}+\frac{793}{333}a^{2}+\frac{400}{333}a-\frac{149}{333}$ | 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: | \( 404.15700628 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 404.15700628 \cdot 2}{18\cdot\sqrt{32357746661769}}\cr\approx \mathstrut & 0.48573307559 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 6.4.5688387.1, 6.2.1896129.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.4.5688387.1 |
Degree 8 sibling: | 8.0.4931831529.2 |
Degree 12 sibling: | 12.4.9351388785251241.1 |
Minimal sibling: | 6.4.5688387.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |