Normalized defining polynomial
\( x^{12} - 5 x^{11} + 15 x^{10} - 27 x^{9} + 38 x^{8} - 39 x^{7} + 39 x^{6} - 26 x^{5} + 21 x^{4} + \cdots + 16 \)
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)
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Discriminant: | \(5231045973904\) \(\medspace = 2^{4}\cdot 83^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.48\) | 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: | $2\cdot 83^{1/2}\approx 18.2208671582886$ | ||
Ramified primes: | \(2\), \(83\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2392}a^{11}+\frac{175}{2392}a^{10}+\frac{419}{2392}a^{9}-\frac{1151}{2392}a^{8}+\frac{37}{92}a^{7}+\frac{3}{8}a^{6}-\frac{89}{184}a^{5}-\frac{7}{92}a^{4}+\frac{749}{2392}a^{3}+\frac{427}{1196}a^{2}+\frac{165}{598}a-\frac{103}{299}$
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)
| |
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: | $\frac{99}{598}a^{11}-\frac{615}{598}a^{10}+\frac{857}{299}a^{9}-\frac{1510}{299}a^{8}+\frac{265}{46}a^{7}-6a^{6}+\frac{205}{46}a^{5}-\frac{167}{46}a^{4}+\frac{149}{299}a^{3}-\frac{1082}{299}a^{2}+\frac{2251}{598}a-\frac{1320}{299}$, $\frac{823}{2392}a^{11}-\frac{3083}{2392}a^{10}+\frac{7565}{2392}a^{9}-\frac{9609}{2392}a^{8}+\frac{459}{92}a^{7}-\frac{31}{8}a^{6}+\frac{813}{184}a^{5}-\frac{57}{92}a^{4}+\frac{5271}{2392}a^{3}-\frac{3193}{1196}a^{2}+\frac{1370}{299}a-\frac{451}{299}$, $\frac{139}{1196}a^{11}-\frac{193}{1196}a^{10}+\frac{235}{1196}a^{9}+\frac{873}{1196}a^{8}-\frac{16}{23}a^{7}+\frac{9}{4}a^{6}-\frac{89}{92}a^{5}+\frac{77}{23}a^{4}+\frac{1255}{1196}a^{3}+\frac{524}{299}a^{2}+\frac{211}{299}a+\frac{967}{299}$, $\frac{343}{2392}a^{11}-\frac{2167}{2392}a^{10}+\frac{6177}{2392}a^{9}-\frac{10877}{2392}a^{8}+\frac{501}{92}a^{7}-\frac{47}{8}a^{6}+\frac{937}{184}a^{5}-\frac{331}{92}a^{4}+\frac{2159}{2392}a^{3}-\frac{3039}{1196}a^{2}+\frac{939}{299}a-\frac{944}{299}$, $\frac{1281}{2392}a^{11}-\frac{4859}{2392}a^{10}+\frac{12293}{2392}a^{9}-\frac{15909}{2392}a^{8}+\frac{194}{23}a^{7}-\frac{49}{8}a^{6}+\frac{1405}{184}a^{5}-\frac{33}{46}a^{4}+\frac{11041}{2392}a^{3}-\frac{1437}{598}a^{2}+\frac{2378}{299}a-\frac{682}{299}$ | 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: | \( 23.2959616796 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 23.2959616796 \cdot 1}{2\cdot\sqrt{5231045973904}}\cr\approx \mathstrut & 0.313354249925 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T21):
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
\(\Q(\sqrt{-83}) \), 3.1.83.1 x3, 6.2.2287148.1, 6.0.27556.1, 6.0.571787.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.2287148.1, 6.0.27556.1 |
Degree 8 siblings: | 8.4.12149330176.1, 8.0.1763584.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.27556.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(83\) | 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |