Normalized defining polynomial
\( x^{12} - 2x^{9} + 4x^{8} - 4x^{7} + x^{6} + 4x^{4} - 8x^{3} + 8x^{2} - 4x + 1 \)
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: | \(46911328256\) \(\medspace = 2^{12}\cdot 13^{4}\cdot 401\) | 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: | \(7.75\) | 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: | $2\cdot 13^{2/3}401^{1/2}\approx 221.4272587292568$ | ||
Ramified primes: | \(2\), \(13\), \(401\) | 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: | \(\Q(\sqrt{401}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{83}a^{11}-\frac{2}{83}a^{10}+\frac{4}{83}a^{9}-\frac{10}{83}a^{8}+\frac{24}{83}a^{7}+\frac{31}{83}a^{6}+\frac{22}{83}a^{5}+\frac{39}{83}a^{4}+\frac{9}{83}a^{3}-\frac{26}{83}a^{2}-\frac{23}{83}a-\frac{41}{83}$
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: | \( -\frac{82}{83} a^{11} - \frac{2}{83} a^{10} + \frac{4}{83} a^{9} + \frac{156}{83} a^{8} - \frac{308}{83} a^{7} + \frac{363}{83} a^{6} + \frac{22}{83} a^{5} - \frac{44}{83} a^{4} - \frac{323}{83} a^{3} + \frac{638}{83} a^{2} - \frac{604}{83} a + \frac{208}{83} \) (order $4$) | 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: | $\frac{102}{83}a^{11}-\frac{38}{83}a^{10}-\frac{90}{83}a^{9}-\frac{273}{83}a^{8}+\frac{456}{83}a^{7}-\frac{407}{83}a^{6}+\frac{3}{83}a^{5}+\frac{77}{83}a^{4}+\frac{503}{83}a^{3}-\frac{992}{83}a^{2}+\frac{725}{83}a-\frac{281}{83}$, $\frac{169}{83}a^{11}+\frac{77}{83}a^{10}+\frac{12}{83}a^{9}-\frac{362}{83}a^{8}+\frac{487}{83}a^{7}-\frac{405}{83}a^{6}-\frac{17}{83}a^{5}+\frac{34}{83}a^{4}+\frac{691}{83}a^{3}-\frac{991}{83}a^{2}+\frac{844}{83}a-\frac{289}{83}$, $\frac{208}{83}a^{11}+\frac{82}{83}a^{10}+\frac{2}{83}a^{9}-\frac{420}{83}a^{8}+\frac{676}{83}a^{7}-\frac{524}{83}a^{6}-\frac{155}{83}a^{5}-\frac{22}{83}a^{4}+\frac{876}{83}a^{3}-\frac{1341}{83}a^{2}+\frac{1026}{83}a-\frac{228}{83}$, $\frac{102}{83}a^{11}+\frac{128}{83}a^{10}+\frac{76}{83}a^{9}-\frac{190}{83}a^{8}+\frac{124}{83}a^{7}-\frac{75}{83}a^{6}-\frac{163}{83}a^{5}-\frac{89}{83}a^{4}+\frac{337}{83}a^{3}-\frac{245}{83}a^{2}+\frac{144}{83}a+\frac{51}{83}$, $\frac{177}{83}a^{11}-\frac{22}{83}a^{10}-\frac{122}{83}a^{9}-\frac{442}{83}a^{8}+\frac{762}{83}a^{7}-\frac{489}{83}a^{6}-\frac{7}{83}a^{5}+\frac{97}{83}a^{4}+\frac{846}{83}a^{3}-\frac{1448}{83}a^{2}+\frac{1075}{83}a-\frac{368}{83}$ | 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: | \( 2.52000020832 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2.52000020832 \cdot 1}{4\cdot\sqrt{46911328256}}\cr\approx \mathstrut & 0.178970285376 \end{aligned}\]
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 $A_4^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 6.0.10816.1 |
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 | R | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(13\) | 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.6.4.1 | $x^{6} + 130 x^{3} - 1521$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(401\) | $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |