Normalized defining polynomial
\( x^{12} + 2x^{10} - 10x^{9} + 10x^{8} + x^{7} + 22x^{6} - 34x^{5} + 24x^{4} - 32x^{3} + 47x^{2} - 66x + 36 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(513710701744969\) \(\medspace = 283^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.82\) | 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: | $283^{1/2}\approx 16.822603841260722$ | ||
Ramified primes: | \(283\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{21}a^{9}-\frac{1}{3}a^{8}-\frac{2}{21}a^{7}+\frac{1}{7}a^{6}-\frac{10}{21}a^{5}+\frac{5}{21}a^{4}+\frac{10}{21}a^{3}-\frac{3}{7}a^{2}+\frac{1}{3}a-\frac{2}{7}$, $\frac{1}{63}a^{10}-\frac{1}{63}a^{9}-\frac{23}{63}a^{8}-\frac{10}{21}a^{7}-\frac{13}{63}a^{6}-\frac{13}{63}a^{5}-\frac{23}{63}a^{4}+\frac{1}{7}a^{3}+\frac{16}{63}a^{2}-\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{1044918}a^{11}-\frac{355}{174153}a^{10}-\frac{1277}{174153}a^{9}-\frac{16627}{74637}a^{8}-\frac{68711}{522459}a^{7}-\frac{17453}{1044918}a^{6}-\frac{26406}{58051}a^{5}+\frac{172436}{522459}a^{4}+\frac{41474}{522459}a^{3}-\frac{68944}{522459}a^{2}+\frac{70739}{348306}a-\frac{28788}{58051}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
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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)
| |
Fundamental units: | $\frac{11021}{116102}a^{11}+\frac{41414}{522459}a^{10}+\frac{141475}{522459}a^{9}-\frac{362266}{522459}a^{8}+\frac{67264}{174153}a^{7}+\frac{49979}{149274}a^{6}+\frac{166924}{74637}a^{5}-\frac{466873}{522459}a^{4}+\frac{107482}{58051}a^{3}-\frac{750349}{522459}a^{2}+\frac{283901}{116102}a-\frac{206204}{58051}$, $\frac{5270}{58051}a^{11}+\frac{49184}{522459}a^{10}+\frac{131581}{522459}a^{9}-\frac{363793}{522459}a^{8}+\frac{24664}{174153}a^{7}+\frac{235381}{522459}a^{6}+\frac{1375258}{522459}a^{5}-\frac{67543}{74637}a^{4}+\frac{11930}{58051}a^{3}-\frac{1038805}{522459}a^{2}+\frac{180349}{58051}a-\frac{185277}{58051}$, $\frac{21955}{149274}a^{11}+\frac{70760}{522459}a^{10}+\frac{201625}{522459}a^{9}-\frac{594224}{522459}a^{8}+\frac{190789}{522459}a^{7}+\frac{40615}{49758}a^{6}+\frac{2033242}{522459}a^{5}-\frac{864650}{522459}a^{4}+\frac{379712}{522459}a^{3}-\frac{1666274}{522459}a^{2}+\frac{1539809}{348306}a-\frac{225363}{58051}$, $\frac{29611}{1044918}a^{11}+\frac{517}{24879}a^{10}+\frac{11126}{174153}a^{9}-\frac{130720}{522459}a^{8}+\frac{77836}{522459}a^{7}+\frac{184837}{1044918}a^{6}+\frac{174763}{174153}a^{5}-\frac{66016}{74637}a^{4}+\frac{382601}{522459}a^{3}-\frac{278350}{522459}a^{2}+\frac{76763}{49758}a-\frac{86928}{58051}$, $\frac{20182}{522459}a^{11}+\frac{2834}{74637}a^{10}+\frac{71431}{522459}a^{9}-\frac{43945}{174153}a^{8}+\frac{84995}{522459}a^{7}-\frac{16150}{522459}a^{6}+\frac{510484}{522459}a^{5}-\frac{8336}{58051}a^{4}+\frac{694996}{522459}a^{3}-\frac{40960}{174153}a^{2}+\frac{53021}{24879}a-\frac{108067}{58051}$ | 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: | \( 79.9490476458 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 79.9490476458 \cdot 4}{2\cdot\sqrt{513710701744969}}\cr\approx \mathstrut & 0.434073420914 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{-283}) \), 3.1.283.1 x3, 6.0.22665187.2, 6.2.80089.1, 6.0.22665187.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.2.283.1 |
Degree 6 siblings: | 6.2.80089.1, 6.0.22665187.2 |
Degree 8 sibling: | 8.0.6414247921.2 |
Degree 12 sibling: | 12.2.1815232161643.1 |
Minimal sibling: | 4.2.283.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(283\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |