Normalized defining polynomial
\( x^{12} - 2x^{11} + 3x^{10} - 2x^{9} + 4x^{6} - 11x^{5} + 17x^{4} - 16x^{3} + 11x^{2} - 5x + 1 \)
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: | \(69637938597\) \(\medspace = 3^{6}\cdot 19^{4}\cdot 733\) | 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: | \(8.01\) | 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: | $3^{1/2}19^{2/3}733^{1/2}\approx 333.8989207608368$ | ||
Ramified primes: | \(3\), \(19\), \(733\) | 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: | \(\Q(\sqrt{733}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11}a^{11}+\frac{4}{11}a^{10}+\frac{5}{11}a^{9}-\frac{5}{11}a^{8}+\frac{3}{11}a^{7}-\frac{4}{11}a^{6}+\frac{2}{11}a^{5}+\frac{1}{11}a^{4}+\frac{1}{11}a^{3}+\frac{1}{11}a^{2}-\frac{5}{11}a-\frac{2}{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{39}{11} a^{11} - \frac{53}{11} a^{10} + \frac{85}{11} a^{9} - \frac{30}{11} a^{8} - \frac{15}{11} a^{7} - \frac{13}{11} a^{6} + \frac{144}{11} a^{5} - \frac{335}{11} a^{4} + \frac{457}{11} a^{3} - \frac{357}{11} a^{2} + \frac{234}{11} a - \frac{67}{11} \) (order $6$) | 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: | $\frac{21}{11}a^{11}-\frac{26}{11}a^{10}+\frac{39}{11}a^{9}-\frac{6}{11}a^{8}-\frac{14}{11}a^{7}-\frac{7}{11}a^{6}+\frac{86}{11}a^{5}-\frac{166}{11}a^{4}+\frac{208}{11}a^{3}-\frac{133}{11}a^{2}+\frac{82}{11}a-\frac{20}{11}$, $a^{11}-2a^{10}+3a^{9}-2a^{8}+4a^{5}-11a^{4}+17a^{3}-16a^{2}+11a-5$, $\frac{58}{11}a^{11}-\frac{87}{11}a^{10}+\frac{125}{11}a^{9}-\frac{37}{11}a^{8}-\frac{35}{11}a^{7}-\frac{12}{11}a^{6}+\frac{237}{11}a^{5}-\frac{525}{11}a^{4}+\frac{685}{11}a^{3}-\frac{503}{11}a^{2}+\frac{293}{11}a-\frac{83}{11}$, $\frac{20}{11}a^{11}-\frac{30}{11}a^{10}+\frac{45}{11}a^{9}-\frac{23}{11}a^{8}-\frac{6}{11}a^{7}-\frac{3}{11}a^{6}+\frac{73}{11}a^{5}-\frac{178}{11}a^{4}+\frac{262}{11}a^{3}-\frac{211}{11}a^{2}+\frac{142}{11}a-\frac{40}{11}$, $\frac{19}{11}a^{11}-\frac{23}{11}a^{10}+\frac{29}{11}a^{9}-\frac{7}{11}a^{8}-\frac{9}{11}a^{7}-\frac{21}{11}a^{6}+\frac{71}{11}a^{5}-\frac{135}{11}a^{4}+\frac{173}{11}a^{3}-\frac{124}{11}a^{2}+\frac{92}{11}a-\frac{38}{11}$ | 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: | \( 4.89240598709 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4.89240598709 \cdot 1}{6\cdot\sqrt{69637938597}}\cr\approx \mathstrut & 0.190119864172 \end{aligned}\]
Galois group
$A_4^2:D_4$ (as 12T208):
A solvable group of order 1152 |
The 44 conjugacy class representatives for $A_4^2:D_4$ |
Character table for $A_4^2:D_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 6.0.9747.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 36 siblings: | 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 | ${\href{/padicField/2.12.0.1}{12} }$ | R | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(19\) | 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(733\) | $\Q_{733}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{733}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{733}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{733}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |