Normalized defining polynomial
\( x^{12} - 2 x^{11} - 10 x^{10} + 24 x^{9} + 27 x^{8} - 106 x^{7} + 22 x^{6} + 222 x^{5} - 287 x^{4} + \cdots + 526 \)
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: | \(1339147769319424\) \(\medspace = 2^{12}\cdot 83^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.22\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2988158496580}a^{11}+\frac{98604251073}{597631699316}a^{10}-\frac{106779345455}{597631699316}a^{9}-\frac{130004083101}{2988158496580}a^{8}-\frac{183809811}{3474602903}a^{7}+\frac{529826155447}{1494079248290}a^{6}-\frac{35282236057}{149407924829}a^{5}+\frac{568910268941}{1494079248290}a^{4}+\frac{14188142829}{69492058060}a^{3}+\frac{283907429599}{597631699316}a^{2}+\frac{335357640984}{747039624145}a-\frac{447393549649}{1494079248290}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{39979624527}{2988158496580}a^{11}-\frac{17570448781}{597631699316}a^{10}-\frac{82918291569}{597631699316}a^{9}+\frac{1140391924033}{2988158496580}a^{8}+\frac{1222342630}{3474602903}a^{7}-\frac{2749710032261}{1494079248290}a^{6}+\frac{135565804969}{149407924829}a^{5}+\frac{5499778066407}{1494079248290}a^{4}-\frac{456931364997}{69492058060}a^{3}-\frac{2028101336751}{597631699316}a^{2}+\frac{10857385169368}{747039624145}a+\frac{17462287123387}{1494079248290}$, $\frac{16223203961}{2988158496580}a^{11}-\frac{7315755423}{597631699316}a^{10}-\frac{34866539119}{597631699316}a^{9}+\frac{523476295939}{2988158496580}a^{8}+\frac{368846666}{3474602903}a^{7}-\frac{1227938493083}{1494079248290}a^{6}+\frac{100067053789}{149407924829}a^{5}+\frac{2038198858541}{1494079248290}a^{4}-\frac{226470462011}{69492058060}a^{3}-\frac{272313108289}{597631699316}a^{2}+\frac{4451364059424}{747039624145}a+\frac{4626612054041}{1494079248290}$, $\frac{91998397}{2988158496580}a^{11}-\frac{1233355223}{597631699316}a^{10}+\frac{11086862929}{597631699316}a^{9}-\frac{114856910917}{2988158496580}a^{8}-\frac{132743858}{3474602903}a^{7}+\frac{342862469269}{1494079248290}a^{6}-\frac{29944625766}{149407924829}a^{5}-\frac{385805396543}{1494079248290}a^{4}+\frac{44579840413}{69492058060}a^{3}-\frac{41645359349}{597631699316}a^{2}-\frac{1300345045577}{747039624145}a-\frac{1114493421523}{1494079248290}$, $\frac{220520194397}{2988158496580}a^{11}-\frac{139420347547}{597631699316}a^{10}-\frac{283617022879}{597631699316}a^{9}+\frac{7026427115463}{2988158496580}a^{8}-\frac{2416241283}{3474602903}a^{7}-\frac{10874069066841}{1494079248290}a^{6}+\frac{1525766054544}{149407924829}a^{5}+\frac{7538779174577}{1494079248290}a^{4}-\frac{1968135147407}{69492058060}a^{3}+\frac{2546037065079}{597631699316}a^{2}+\frac{40586920777563}{747039624145}a+\frac{47846595172397}{1494079248290}$, $\frac{479901578857}{2988158496580}a^{11}-\frac{326717863459}{597631699316}a^{10}-\frac{530217160439}{597631699316}a^{9}+\frac{15868200104843}{2988158496580}a^{8}-\frac{10599846091}{3474602903}a^{7}-\frac{21435298015921}{1494079248290}a^{6}+\frac{3870335840665}{149407924829}a^{5}+\frac{2320806962437}{1494079248290}a^{4}-\frac{3982451331167}{69492058060}a^{3}+\frac{14066589084623}{597631699316}a^{2}+\frac{76655081207578}{747039624145}a+\frac{70378168356117}{1494079248290}$ | 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: | \( 265.254382815 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 265.254382815 \cdot 4}{2\cdot\sqrt{1339147769319424}}\cr\approx \mathstrut & 0.891984942599 \end{aligned}\]
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
\(\Q(\sqrt{83}) \), 3.1.83.1, 6.2.36594368.1, 6.0.27556.1, 6.0.9148592.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(83\) | 83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |