Normalized defining polynomial
\( x^{12} - x^{11} - 4x^{10} + 8x^{9} - 7x^{8} - 30x^{7} + 41x^{6} + 50x^{5} - 33x^{4} - 10x^{3} + 10x^{2} - 7x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(181928116000000\) \(\medspace = 2^{8}\cdot 5^{6}\cdot 19^{4}\cdot 349\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.43\) | 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))
| |
Galois root discriminant: | $2^{2/3}5^{1/2}19^{2/3}349^{1/2}\approx 472.1573776028913$ | ||
Ramified primes: | \(2\), \(5\), \(19\), \(349\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{349}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{331}a^{11}+\frac{65}{331}a^{10}-\frac{17}{331}a^{9}-\frac{121}{331}a^{8}-\frac{49}{331}a^{7}+\frac{46}{331}a^{6}+\frac{98}{331}a^{5}-\frac{102}{331}a^{4}-\frac{145}{331}a^{3}+\frac{19}{331}a^{2}-\frac{60}{331}a+\frac{5}{331}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{1461}{331}a^{11}-\frac{3342}{331}a^{10}-\frac{1336}{331}a^{9}+\frac{12882}{331}a^{8}-\frac{26904}{331}a^{7}-\frac{7269}{331}a^{6}+\frac{65062}{331}a^{5}-\frac{10995}{331}a^{4}-\frac{24168}{331}a^{3}+\frac{12864}{331}a^{2}-\frac{6565}{331}a+\frac{354}{331}$, $\frac{900}{331}a^{11}-\frac{2404}{331}a^{10}-\frac{74}{331}a^{9}+\frac{8274}{331}a^{8}-\frac{19606}{331}a^{7}+\frac{1680}{331}a^{6}+\frac{42191}{331}a^{5}-\frac{21628}{331}a^{4}-\frac{12333}{331}a^{3}+\frac{14121}{331}a^{2}-\frac{6998}{331}a+\frac{1521}{331}$, $\frac{1314}{331}a^{11}-\frac{3298}{331}a^{10}-\frac{492}{331}a^{9}+\frac{11802}{331}a^{8}-\frac{26983}{331}a^{7}-\frac{791}{331}a^{6}+\frac{59593}{331}a^{5}-\frac{24136}{331}a^{4}-\frac{17417}{331}a^{3}+\frac{19008}{331}a^{2}-\frac{8999}{331}a+\frac{943}{331}$, $\frac{360}{331}a^{11}-\frac{763}{331}a^{10}-\frac{493}{331}a^{9}+\frac{3111}{331}a^{8}-\frac{6055}{331}a^{7}-\frac{2969}{331}a^{6}+\frac{15751}{331}a^{5}+\frac{683}{331}a^{4}-\frac{6191}{331}a^{3}+\frac{1544}{331}a^{2}-\frac{1078}{331}a-\frac{186}{331}$, $\frac{414}{331}a^{11}-\frac{894}{331}a^{10}-\frac{418}{331}a^{9}+\frac{3528}{331}a^{8}-\frac{7377}{331}a^{7}-\frac{2471}{331}a^{6}+\frac{17402}{331}a^{5}-\frac{2508}{331}a^{4}-\frac{5084}{331}a^{3}+\frac{4556}{331}a^{2}-\frac{1670}{331}a+\frac{415}{331}$, $a$, $\frac{999}{331}a^{11}-\frac{2258}{331}a^{10}-\frac{764}{331}a^{9}+\frac{8542}{331}a^{8}-\frac{18499}{331}a^{7}-\frac{4027}{331}a^{6}+\frac{41963}{331}a^{5}-\frac{9549}{331}a^{4}-\frac{11793}{331}a^{3}+\frac{10044}{331}a^{2}-\frac{6318}{331}a+\frac{692}{331}$, $\frac{914}{331}a^{11}-\frac{2156}{331}a^{10}-\frac{643}{331}a^{9}+\frac{7904}{331}a^{8}-\frac{17313}{331}a^{7}-\frac{2972}{331}a^{6}+\frac{39591}{331}a^{5}-\frac{8492}{331}a^{4}-\frac{12046}{331}a^{3}+\frac{8098}{331}a^{2}-\frac{3866}{331}a+\frac{267}{331}$, $\frac{627}{331}a^{11}-\frac{1613}{331}a^{10}-\frac{67}{331}a^{9}+\frac{5559}{331}a^{8}-\frac{13180}{331}a^{7}+\frac{707}{331}a^{6}+\frac{27684}{331}a^{5}-\frac{13642}{331}a^{4}-\frac{7503}{331}a^{3}+\frac{8934}{331}a^{2}-\frac{4520}{331}a+\frac{818}{331}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 493.218202324 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 493.218202324 \cdot 1}{2\cdot\sqrt{181928116000000}}\cr\approx \mathstrut & 0.184781671470 \end{aligned}\]
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 | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{12} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{3}^{12}$ |
\(5\) | 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(19\) | 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(349\) | $\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |