Normalized defining polynomial
\( x^{12} - x^{10} - 2x^{9} - 5x^{8} + 12x^{7} - 15x^{6} + 26x^{5} - 8x^{4} - 4x^{3} - 20x^{2} + 16x - 4 \)
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: | \(-114516604000000\) \(\medspace = -\,2^{8}\cdot 5^{6}\cdot 31^{5}\) | 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: | \(14.84\) | 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: | $2^{2/3}5^{1/2}31^{1/2}\approx 19.762983718745776$ | ||
Ramified primes: | \(2\), \(5\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{20}a^{10}-\frac{1}{10}a^{9}+\frac{1}{10}a^{8}+\frac{1}{20}a^{7}-\frac{1}{5}a^{6}-\frac{1}{20}a^{5}-\frac{9}{20}a^{4}-\frac{1}{5}a^{2}-\frac{3}{10}a-\frac{1}{5}$, $\frac{1}{4120}a^{11}-\frac{71}{4120}a^{10}-\frac{11}{412}a^{9}-\frac{54}{515}a^{8}-\frac{233}{4120}a^{7}-\frac{191}{824}a^{6}+\frac{187}{412}a^{5}-\frac{967}{2060}a^{4}-\frac{218}{515}a^{3}+\frac{11}{206}a^{2}+\frac{21}{103}a+\frac{29}{1030}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{271}{4120}a^{11}+\frac{329}{4120}a^{10}+\frac{3}{206}a^{9}-\frac{341}{2060}a^{8}-\frac{2373}{4120}a^{7}+\frac{151}{824}a^{6}-\frac{205}{412}a^{5}+\frac{2653}{2060}a^{4}+\frac{809}{1030}a^{3}+\frac{97}{206}a^{2}-\frac{77}{103}a-\frac{381}{1030}$, $\frac{573}{2060}a^{11}+\frac{52}{515}a^{10}-\frac{153}{515}a^{9}-\frac{1469}{2060}a^{8}-\frac{3523}{2060}a^{7}+\frac{2793}{1030}a^{6}-\frac{1519}{515}a^{5}+\frac{12149}{2060}a^{4}+\frac{462}{515}a^{3}-\frac{1757}{1030}a^{2}-\frac{5613}{1030}a+\frac{961}{515}$, $\frac{399}{4120}a^{11}+\frac{61}{824}a^{10}-\frac{109}{2060}a^{9}-\frac{77}{412}a^{8}-\frac{2533}{4120}a^{7}+\frac{2939}{4120}a^{6}-\frac{438}{515}a^{5}+\frac{2889}{2060}a^{4}+\frac{621}{1030}a^{3}-\frac{512}{515}a^{2}-\frac{438}{515}a+\frac{447}{1030}$, $\frac{139}{4120}a^{11}+\frac{431}{4120}a^{10}+\frac{4}{103}a^{9}-\frac{77}{1030}a^{8}-\frac{1487}{4120}a^{7}-\frac{181}{824}a^{6}+\frac{35}{103}a^{5}-\frac{513}{2060}a^{4}+\frac{1113}{515}a^{3}-\frac{8}{103}a^{2}+\frac{35}{103}a-\frac{1119}{1030}$, $\frac{267}{4120}a^{11}+\frac{407}{4120}a^{10}-\frac{59}{2060}a^{9}-\frac{99}{1030}a^{8}-\frac{1647}{4120}a^{7}+\frac{1279}{4120}a^{6}-\frac{27}{2060}a^{5}-\frac{277}{2060}a^{4}+\frac{1019}{515}a^{3}-\frac{1589}{1030}a^{2}-\frac{393}{515}a+\frac{739}{1030}$, $\frac{23}{824}a^{11}+\frac{281}{4120}a^{10}+\frac{41}{515}a^{9}+\frac{43}{1030}a^{8}-\frac{839}{4120}a^{7}-\frac{439}{4120}a^{6}-\frac{182}{515}a^{5}-\frac{377}{2060}a^{4}+\frac{169}{206}a^{3}+\frac{227}{515}a^{2}+\frac{333}{515}a+\frac{39}{1030}$ | 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: | \( 210.935703626 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 210.935703626 \cdot 1}{2\cdot\sqrt{114516604000000}}\cr\approx \mathstrut & 0.386051463505 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 9 conjugacy class representatives for $D_{12}$ |
Character table for $D_{12}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 3.1.620.1, 4.2.775.1, 6.2.1922000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.0.710002944800000.1 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(31\) | 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |