Normalized defining polynomial
\( x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 108 x^{8} - 198 x^{7} + 288 x^{6} - 315 x^{5} + 315 x^{4} + \cdots + 125 \)
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: | \(1315127813325481\) \(\medspace = 331^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(18.19\) | 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: | $331^{1/2}\approx 18.193405398660254$ | ||
Ramified primes: | \(331\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5050}a^{10}-\frac{1}{1010}a^{9}-\frac{199}{5050}a^{8}+\frac{413}{2525}a^{7}-\frac{1731}{5050}a^{6}+\frac{2281}{5050}a^{5}+\frac{517}{2525}a^{4}+\frac{77}{2525}a^{3}+\frac{359}{5050}a^{2}+\frac{233}{505}a+\frac{47}{202}$, $\frac{1}{3014850}a^{11}+\frac{293}{3014850}a^{10}+\frac{9421}{3014850}a^{9}+\frac{32169}{502475}a^{8}+\frac{105039}{1004950}a^{7}-\frac{376889}{1004950}a^{6}+\frac{185172}{502475}a^{5}+\frac{209446}{502475}a^{4}+\frac{411337}{1004950}a^{3}-\frac{318034}{1507425}a^{2}+\frac{41183}{200990}a-\frac{22691}{60297}$
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)
| |
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: | $\frac{13391}{1507425}a^{11}-\frac{99023}{1507425}a^{10}+\frac{350801}{1507425}a^{9}-\frac{2749}{4975}a^{8}+\frac{555087}{502475}a^{7}-\frac{1067047}{502475}a^{6}+\frac{1515277}{502475}a^{5}-\frac{1476356}{502475}a^{4}+\frac{1212739}{502475}a^{3}-\frac{4549867}{1507425}a^{2}+\frac{262498}{100495}a-\frac{127940}{60297}$, $\frac{33202}{1507425}a^{11}-\frac{314477}{3014850}a^{10}+\frac{882329}{3014850}a^{9}-\frac{539291}{1004950}a^{8}+\frac{549069}{502475}a^{7}-\frac{1682803}{1004950}a^{6}+\frac{1406023}{1004950}a^{5}-\frac{148162}{502475}a^{4}+\frac{290018}{502475}a^{3}+\frac{1017077}{3014850}a^{2}+\frac{70756}{100495}a+\frac{44599}{120594}$, $\frac{15531}{502475}a^{11}-\frac{170841}{1004950}a^{10}+\frac{527977}{1004950}a^{9}-\frac{1094589}{1004950}a^{8}+\frac{1104156}{502475}a^{7}-\frac{3816897}{1004950}a^{6}+\frac{4436577}{1004950}a^{5}-\frac{1624518}{502475}a^{4}+\frac{1502757}{502475}a^{3}-\frac{2694219}{1004950}a^{2}+\frac{333254}{100495}a-\frac{30229}{40198}$, $\frac{44927}{3014850}a^{11}-\frac{181906}{1507425}a^{10}+\frac{641371}{1507425}a^{9}-\frac{206021}{200990}a^{8}+\frac{2048067}{1004950}a^{7}-\frac{2015971}{502475}a^{6}+\frac{5629077}{1004950}a^{5}-\frac{545027}{100495}a^{4}+\frac{989669}{200990}a^{3}-\frac{16374493}{3014850}a^{2}+\frac{648613}{200990}a-\frac{494045}{120594}$, $\frac{833}{3014850}a^{11}-\frac{27514}{1507425}a^{10}+\frac{149314}{1507425}a^{9}-\frac{13187}{40198}a^{8}+\frac{724333}{1004950}a^{7}-\frac{740264}{502475}a^{6}+\frac{2496043}{1004950}a^{5}-\frac{317834}{100495}a^{4}+\frac{498047}{200990}a^{3}-\frac{3498997}{3014850}a^{2}+\frac{25699}{200990}a-\frac{5345}{120594}$ | 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: | \( 119.503369717 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 119.503369717 \cdot 4}{2\cdot\sqrt{1315127813325481}}\cr\approx \mathstrut & 0.405513555181 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{-331}) \), 3.1.331.1 x3, 6.0.36264691.2, 6.2.109561.1, 6.0.36264691.1 |
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: | 4.2.331.1 |
Degree 6 siblings: | 6.2.109561.1, 6.0.36264691.2 |
Degree 8 sibling: | 8.0.12003612721.2 |
Degree 12 sibling: | 12.2.3973195810651.1 |
Minimal sibling: | 4.2.331.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(331\) | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |