Normalized defining polynomial
\( x^{12} - 6 x^{11} + 16 x^{10} - 16 x^{9} - 16 x^{8} + 64 x^{7} - 70 x^{6} + 22 x^{5} + 15 x^{4} + \cdots - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-7504960159744\) \(\medspace = -\,2^{18}\cdot 31^{5}\) | 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: | \(11.83\) | 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: | $2^{7/4}31^{1/2}\approx 18.727652374724588$ | ||
Ramified primes: | \(2\), \(31\) | 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: | \(\Q(\sqrt{-31}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{33}a^{9}-\frac{1}{11}a^{8}+\frac{3}{11}a^{7}+\frac{1}{11}a^{6}+\frac{2}{11}a^{5}+\frac{5}{11}a^{4}+\frac{1}{11}a^{3}-\frac{2}{11}a^{2}-\frac{5}{33}a+\frac{5}{33}$, $\frac{1}{33}a^{10}-\frac{1}{11}a^{7}+\frac{5}{11}a^{6}+\frac{5}{11}a^{4}+\frac{1}{11}a^{3}+\frac{10}{33}a^{2}-\frac{10}{33}a+\frac{5}{11}$, $\frac{1}{33}a^{11}-\frac{1}{11}a^{8}+\frac{5}{11}a^{7}+\frac{5}{11}a^{5}+\frac{1}{11}a^{4}+\frac{10}{33}a^{3}-\frac{10}{33}a^{2}+\frac{5}{11}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{3}{11}a^{11}-\frac{50}{33}a^{10}+\frac{128}{33}a^{9}-\frac{38}{11}a^{8}-\frac{49}{11}a^{7}+\frac{175}{11}a^{6}-\frac{183}{11}a^{5}+\frac{47}{11}a^{4}+\frac{64}{11}a^{3}-\frac{104}{33}a^{2}-\frac{38}{33}a+\frac{2}{3}$, $\frac{2}{11}a^{10}-\frac{10}{11}a^{9}+\frac{68}{33}a^{8}-\frac{35}{33}a^{7}-\frac{11}{3}a^{6}+\frac{271}{33}a^{5}-\frac{184}{33}a^{4}-\frac{17}{33}a^{3}+\frac{53}{33}a^{2}+\frac{13}{33}a-\frac{5}{33}$, $\frac{4}{33}a^{11}-\frac{29}{33}a^{10}+\frac{91}{33}a^{9}-\frac{131}{33}a^{8}-\frac{13}{33}a^{7}+\frac{355}{33}a^{6}-\frac{571}{33}a^{5}+\frac{337}{33}a^{4}+\frac{24}{11}a^{3}-\frac{161}{33}a^{2}-\frac{28}{33}a+\frac{64}{33}$, $\frac{7}{11}a^{11}-\frac{40}{11}a^{10}+\frac{296}{33}a^{9}-\frac{75}{11}a^{8}-\frac{152}{11}a^{7}+\frac{411}{11}a^{6}-\frac{337}{11}a^{5}-\frac{12}{11}a^{4}+\frac{147}{11}a^{3}+\frac{5}{11}a^{2}-\frac{163}{33}a+\frac{10}{33}$, $\frac{5}{33}a^{11}-\frac{10}{11}a^{10}+\frac{74}{33}a^{9}-\frac{50}{33}a^{8}-\frac{148}{33}a^{7}+\frac{355}{33}a^{6}-\frac{229}{33}a^{5}-\frac{161}{33}a^{4}+\frac{259}{33}a^{3}-\frac{13}{33}a^{2}-\frac{116}{33}a-\frac{1}{11}$, $\frac{10}{33}a^{11}-\frac{56}{33}a^{10}+\frac{47}{11}a^{9}-\frac{41}{11}a^{8}-\frac{54}{11}a^{7}+\frac{180}{11}a^{6}-\frac{185}{11}a^{5}+\frac{61}{11}a^{4}+\frac{58}{33}a^{3}+\frac{4}{11}a^{2}-\frac{61}{33}a-\frac{1}{11}$ | 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: | \( 33.2988142487 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 33.2988142487 \cdot 1}{2\cdot\sqrt{7504960159744}}\cr\approx \mathstrut & 0.238058619447 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T22):
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
3.1.31.1, 6.2.61504.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.61504.1, 6.2.1906624.4 |
Degree 8 siblings: | 8.4.15130968064.1, 8.0.15745024.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.61504.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.31 | $x^{12} - 4 x^{11} + 22 x^{10} + 16 x^{9} + 50 x^{8} + 32 x^{7} + 144 x^{6} + 96 x^{5} + 236 x^{4} + 248 x^{2} + 248$ | $4$ | $3$ | $18$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |