Normalized defining polynomial
\( x^{12} - x^{11} + 7 x^{10} + x^{9} + 13 x^{8} + 26 x^{7} - 17 x^{6} + 45 x^{5} - 29 x^{4} + 54 x^{3} + \cdots + 279 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1572266908616041\) \(\medspace = 11^{6}\cdot 31^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.47\) | 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: | $11^{1/2}31^{1/2}\approx 18.466185312619388$ | ||
Ramified primes: | \(11\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{195}a^{10}+\frac{11}{39}a^{9}-\frac{67}{195}a^{8}+\frac{34}{195}a^{7}-\frac{19}{39}a^{6}-\frac{1}{13}a^{5}+\frac{11}{65}a^{4}+\frac{21}{65}a^{3}+\frac{37}{195}a^{2}-\frac{16}{195}a-\frac{3}{65}$, $\frac{1}{1479315825}a^{11}+\frac{136366}{54789475}a^{10}-\frac{532857062}{1479315825}a^{9}+\frac{112340716}{295863165}a^{8}-\frac{119274149}{493105275}a^{7}-\frac{2198158}{4551741}a^{6}-\frac{139139864}{493105275}a^{5}-\frac{33099122}{493105275}a^{4}+\frac{356769568}{1479315825}a^{3}-\frac{624395852}{1479315825}a^{2}-\frac{321725171}{1479315825}a-\frac{15851257}{164368425}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{4}$, which has order $4$
Unit group
Rank: | $5$ | 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: | $\frac{447254}{54789475}a^{11}-\frac{1679017}{54789475}a^{10}+\frac{5178002}{54789475}a^{9}-\frac{1639343}{10957895}a^{8}+\frac{9548207}{54789475}a^{7}-\frac{135826}{2191579}a^{6}-\frac{17028618}{54789475}a^{5}+\frac{33493101}{54789475}a^{4}-\frac{32914363}{54789475}a^{3}+\frac{37667627}{54789475}a^{2}+\frac{6183911}{54789475}a-\frac{12107647}{54789475}$, $\frac{27694}{1479315825}a^{11}+\frac{1625327}{164368425}a^{10}-\frac{9497153}{1479315825}a^{9}+\frac{3087194}{59172633}a^{8}+\frac{30276274}{493105275}a^{7}+\frac{1692242}{59172633}a^{6}+\frac{164536984}{493105275}a^{5}-\frac{108516908}{493105275}a^{4}+\frac{276366772}{1479315825}a^{3}-\frac{242943218}{1479315825}a^{2}-\frac{400432034}{1479315825}a+\frac{86340482}{164368425}$, $\frac{4769941}{1479315825}a^{11}-\frac{2717977}{164368425}a^{10}+\frac{54830008}{1479315825}a^{9}-\frac{21197807}{295863165}a^{8}+\frac{21422851}{493105275}a^{7}-\frac{3056377}{59172633}a^{6}-\frac{79923299}{493105275}a^{5}+\frac{199576693}{493105275}a^{4}-\frac{723962552}{1479315825}a^{3}+\frac{809675833}{1479315825}a^{2}-\frac{1241636381}{1479315825}a-\frac{35923564}{12643725}$, $\frac{3804916}{493105275}a^{11}-\frac{3580186}{164368425}a^{10}+\frac{44186758}{493105275}a^{9}-\frac{14029976}{98621055}a^{8}+\frac{15252152}{54789475}a^{7}-\frac{3071257}{19724211}a^{6}-\frac{33535049}{164368425}a^{5}+\frac{123065428}{164368425}a^{4}-\frac{688468847}{493105275}a^{3}+\frac{728466103}{493105275}a^{2}-\frac{153587066}{493105275}a+\frac{23047158}{54789475}$, $\frac{874459}{54789475}a^{11}-\frac{1443297}{54789475}a^{10}+\frac{5157342}{54789475}a^{9}-\frac{141542}{10957895}a^{8}+\frac{4874962}{54789475}a^{7}+\frac{307227}{2191579}a^{6}-\frac{11230678}{54789475}a^{5}+\frac{3996126}{54789475}a^{4}+\frac{702564}{4214575}a^{3}+\frac{43235737}{54789475}a^{2}+\frac{57039171}{54789475}a+\frac{14956873}{54789475}$ | 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: | \( 353.055983844 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 353.055983844 \cdot 4}{2\cdot\sqrt{1572266908616041}}\cr\approx \mathstrut & 1.09569491328 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), 3.1.31.1, 6.2.327701.1, 6.0.1279091.1, 6.0.3604711.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.327701.1, 6.0.10571.1 |
Degree 8 siblings: | 8.4.13521270961.1, 8.0.14070001.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.10571.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |