Normalized defining polynomial
\( x^{12} - x^{11} - 21 x^{10} + 13 x^{9} + 163 x^{8} - 54 x^{7} - 554 x^{6} + 93 x^{5} + 765 x^{4} - 73 x^{3} - 275 x^{2} - 42 x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(131045349590545581\) \(\medspace = 3^{9}\cdot 367^{5}\) | 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: | \(26.70\) | 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: | $3^{3/4}367^{1/2}\approx 43.66907302809774$ | ||
Ramified primes: | \(3\), \(367\) | 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: | \(\Q(\sqrt{1101}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{9}+\frac{1}{6}a^{8}+\frac{5}{12}a^{6}+\frac{1}{6}a^{5}+\frac{1}{12}a^{4}-\frac{1}{12}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{5}{12}$, $\frac{1}{24}a^{11}-\frac{1}{24}a^{9}-\frac{1}{24}a^{7}-\frac{7}{24}a^{6}+\frac{7}{24}a^{5}+\frac{1}{3}a^{4}-\frac{11}{24}a^{3}-\frac{1}{3}a^{2}-\frac{11}{24}a-\frac{3}{8}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | 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)
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Fundamental units: | $\frac{29}{24}a^{11}-\frac{11}{3}a^{10}-\frac{425}{24}a^{9}+\frac{307}{6}a^{8}+\frac{2155}{24}a^{7}-\frac{5767}{24}a^{6}-\frac{1319}{8}a^{5}+414a^{4}+\frac{435}{8}a^{3}-\frac{875}{6}a^{2}-\frac{791}{24}a+\frac{11}{24}$, $\frac{19}{24}a^{11}-\frac{5}{2}a^{10}-\frac{271}{24}a^{9}+35a^{8}+\frac{1301}{24}a^{7}-\frac{3961}{24}a^{6}-\frac{2027}{24}a^{5}+\frac{1703}{6}a^{4}-\frac{197}{24}a^{3}-\frac{277}{3}a^{2}-\frac{185}{24}a+\frac{11}{8}$, $\frac{35}{24}a^{11}-\frac{19}{4}a^{10}-\frac{485}{24}a^{9}+\frac{131}{2}a^{8}+\frac{2257}{24}a^{7}-\frac{7283}{24}a^{6}-\frac{3391}{24}a^{5}+\frac{6113}{12}a^{4}-\frac{343}{24}a^{3}-\frac{470}{3}a^{2}-\frac{565}{24}a+\frac{1}{8}$, $\frac{25}{6}a^{11}-\frac{163}{12}a^{10}-\frac{689}{12}a^{9}+\frac{1115}{6}a^{8}+\frac{1607}{6}a^{7}-\frac{10273}{12}a^{6}-415a^{5}+\frac{5767}{4}a^{4}-\frac{17}{4}a^{3}-\frac{1448}{3}a^{2}-\frac{167}{3}a+\frac{41}{12}$, $\frac{43}{24}a^{11}-\frac{17}{3}a^{10}-\frac{607}{24}a^{9}+\frac{233}{3}a^{8}+\frac{2957}{24}a^{7}-\frac{2871}{8}a^{6}-\frac{5143}{24}a^{5}+\frac{3667}{6}a^{4}+\frac{1319}{24}a^{3}-\frac{445}{2}a^{2}-\frac{303}{8}a+\frac{137}{24}$, $\frac{67}{24}a^{11}-\frac{107}{12}a^{10}-\frac{937}{24}a^{9}+\frac{733}{6}a^{8}+\frac{4469}{24}a^{7}-\frac{4505}{8}a^{6}-\frac{7279}{24}a^{5}+\frac{11381}{12}a^{4}+\frac{677}{24}a^{3}-317a^{2}-\frac{359}{8}a+\frac{59}{24}$, $\frac{11}{8}a^{11}-\frac{14}{3}a^{10}-\frac{147}{8}a^{9}+\frac{191}{3}a^{8}+\frac{653}{8}a^{7}-\frac{7031}{24}a^{6}-\frac{2681}{24}a^{5}+\frac{1480}{3}a^{4}-\frac{779}{24}a^{3}-\frac{497}{3}a^{2}-\frac{283}{24}a+\frac{23}{24}$, $\frac{13}{12}a^{11}-\frac{37}{12}a^{10}-\frac{49}{3}a^{9}+\frac{251}{6}a^{8}+\frac{1073}{12}a^{7}-\frac{381}{2}a^{6}-\frac{2425}{12}a^{5}+\frac{3835}{12}a^{4}+\frac{985}{6}a^{3}-\frac{231}{2}a^{2}-\frac{293}{4}a-\frac{25}{3}$, $\frac{3}{4}a^{11}-\frac{29}{12}a^{10}-\frac{21}{2}a^{9}+\frac{199}{6}a^{8}+\frac{203}{4}a^{7}-\frac{923}{6}a^{6}-\frac{1051}{12}a^{5}+\frac{3169}{12}a^{4}+\frac{127}{6}a^{3}-\frac{302}{3}a^{2}-\frac{185}{12}a+\frac{17}{6}$, $\frac{5}{3}a^{11}-\frac{59}{12}a^{10}-\frac{299}{12}a^{9}+\frac{206}{3}a^{8}+\frac{785}{6}a^{7}-\frac{1293}{4}a^{6}-\frac{782}{3}a^{5}+\frac{6749}{12}a^{4}+\frac{1579}{12}a^{3}-211a^{2}-60a+\frac{7}{12}$, $\frac{9}{4}a^{11}-\frac{41}{6}a^{10}-\frac{131}{4}a^{9}+\frac{563}{6}a^{8}+\frac{673}{4}a^{7}-\frac{5219}{12}a^{6}-\frac{3977}{12}a^{5}+\frac{4489}{6}a^{4}+\frac{2077}{12}a^{3}-\frac{865}{3}a^{2}-\frac{955}{12}a+\frac{23}{12}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 50889.5043205 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 50889.5043205 \cdot 1}{2\cdot\sqrt{131045349590545581}}\cr\approx \mathstrut & 0.287903779490 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
3.3.1101.1, 4.4.9909.1 x2, 6.6.10909809.1 |
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: | 4.4.9909.1 |
Degree 6 siblings: | 6.6.1334633301.2, 6.6.10909809.1 |
Degree 8 sibling: | 8.8.13224881379609.1 |
Degree 12 sibling: | deg 12 |
Minimal sibling: | 4.4.9909.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
\(367\) | $\Q_{367}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{367}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |