Normalized defining polynomial
\( x^{12} - 3x^{11} + 2x^{10} + 2x^{9} - 9x^{8} + 16x^{7} - 19x^{6} + 16x^{5} - 9x^{4} + 2x^{3} + 2x^{2} - 3x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(31798324000000\) \(\medspace = 2^{8}\cdot 5^{6}\cdot 19^{4}\cdot 61\) | 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: | \(13.34\) | 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: | $2^{2/3}5^{1/2}19^{2/3}61^{1/2}\approx 197.39628913634618$ | ||
Ramified primes: | \(2\), \(5\), \(19\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}+\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}$, $\frac{1}{7}a^{11}+\frac{1}{7}a^{9}-\frac{2}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $a^{11}-3a^{10}+2a^{9}+2a^{8}-9a^{7}+16a^{6}-19a^{5}+16a^{4}-9a^{3}+2a^{2}+2a-3$, $\frac{9}{7}a^{11}-\frac{20}{7}a^{10}+\frac{2}{7}a^{9}+\frac{18}{7}a^{8}-\frac{62}{7}a^{7}+\frac{99}{7}a^{6}-\frac{104}{7}a^{5}+\frac{71}{7}a^{4}-4a^{3}-\frac{6}{7}a^{2}+\frac{23}{7}a-\frac{13}{7}$, $a^{11}-3a^{10}+2a^{9}+2a^{8}-9a^{7}+16a^{6}-19a^{5}+16a^{4}-9a^{3}+2a^{2}+2a-2$, $\frac{2}{7}a^{11}-\frac{2}{7}a^{10}-\frac{5}{7}a^{9}+\frac{1}{7}a^{8}+a^{6}-2a^{5}+2a^{4}-\frac{15}{7}a^{3}+\frac{5}{7}a^{2}+\frac{2}{7}a-\frac{9}{7}$, $\frac{3}{7}a^{11}-\frac{5}{7}a^{10}-\frac{4}{7}a^{9}+\frac{10}{7}a^{8}-\frac{17}{7}a^{7}+\frac{11}{7}a^{6}-\frac{10}{7}a^{5}+\frac{4}{7}a^{4}+\frac{6}{7}a^{3}-\frac{12}{7}a^{2}+\frac{10}{7}a-\frac{5}{7}$, $\frac{11}{7}a^{11}-\frac{26}{7}a^{10}+\frac{4}{7}a^{9}+\frac{29}{7}a^{8}-\frac{82}{7}a^{7}+\frac{114}{7}a^{6}-\frac{117}{7}a^{5}+\frac{86}{7}a^{4}-6a^{3}+\frac{9}{7}a^{2}+\frac{11}{7}a-\frac{12}{7}$, $\frac{11}{7}a^{11}-\frac{26}{7}a^{10}+\frac{4}{7}a^{9}+\frac{29}{7}a^{8}-\frac{82}{7}a^{7}+\frac{114}{7}a^{6}-\frac{117}{7}a^{5}+\frac{86}{7}a^{4}-6a^{3}+\frac{9}{7}a^{2}+\frac{11}{7}a-\frac{19}{7}$ | 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: | \( 91.3485453511 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 91.3485453511 \cdot 1}{2\cdot\sqrt{31798324000000}}\cr\approx \mathstrut & 0.201980285303 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), 6.6.722000.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }$ | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.8.2 | $x^{12} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{3}^{12}$ |
\(5\) | 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(19\) | 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(61\) | $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.6.0.1 | $x^{6} + 49 x^{3} + 3 x^{2} + 29 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |