Normalized defining polynomial
\( x^{12} - 4x^{10} + 3x^{8} - 120x^{6} - 149x^{4} - 148x^{2} + 25 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1740631100058763264\) \(\medspace = 2^{26}\cdot 11^{10}\) | 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: | \(33.12\) | 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: | $2^{13/6}11^{5/6}\approx 33.11760293935955$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{28}a^{8}+\frac{3}{14}a^{6}+\frac{3}{14}a^{4}-\frac{5}{14}a^{2}-\frac{3}{28}$, $\frac{1}{140}a^{9}-\frac{1}{140}a^{7}-\frac{1}{4}a^{6}-\frac{3}{28}a^{5}-\frac{1}{4}a^{4}-\frac{9}{28}a^{3}-\frac{1}{4}a^{2}-\frac{6}{35}a-\frac{1}{4}$, $\frac{1}{5320}a^{10}+\frac{39}{5320}a^{8}-\frac{79}{532}a^{6}-\frac{103}{532}a^{4}+\frac{1921}{5320}a^{2}+\frac{151}{1064}$, $\frac{1}{5320}a^{11}+\frac{1}{5320}a^{9}+\frac{289}{2660}a^{7}+\frac{87}{532}a^{5}-\frac{359}{5320}a^{3}-\frac{2323}{5320}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{27}{1330}a^{11}-\frac{33}{380}a^{9}+\frac{121}{1330}a^{7}-\frac{330}{133}a^{5}-\frac{1474}{665}a^{3}-\frac{8877}{2660}a-1$, $\frac{9}{2660}a^{10}-\frac{29}{2660}a^{8}-\frac{4}{133}a^{6}-\frac{13}{38}a^{4}-\frac{1521}{2660}a^{2}-\frac{9}{532}$, $\frac{79}{1330}a^{11}-\frac{43}{190}a^{9}+\frac{157}{1330}a^{7}-\frac{936}{133}a^{5}-\frac{6628}{665}a^{3}-\frac{13467}{1330}a$, $\frac{2}{665}a^{11}+\frac{33}{5320}a^{10}-\frac{3}{266}a^{9}-\frac{233}{5320}a^{8}+\frac{1}{380}a^{7}+\frac{18}{133}a^{6}-\frac{223}{532}a^{5}-\frac{227}{266}a^{4}-\frac{71}{380}a^{3}+\frac{2783}{5320}a^{2}+\frac{283}{2660}a-\frac{223}{1064}$, $\frac{2}{665}a^{11}-\frac{33}{5320}a^{10}-\frac{3}{266}a^{9}+\frac{233}{5320}a^{8}+\frac{1}{380}a^{7}-\frac{18}{133}a^{6}-\frac{223}{532}a^{5}+\frac{227}{266}a^{4}-\frac{71}{380}a^{3}-\frac{2783}{5320}a^{2}+\frac{283}{2660}a+\frac{223}{1064}$, $\frac{363}{2660}a^{11}-\frac{53}{1064}a^{10}-\frac{683}{1330}a^{9}+\frac{25}{152}a^{8}+\frac{369}{1330}a^{7}-\frac{3}{532}a^{6}-\frac{4329}{266}a^{5}+\frac{3165}{532}a^{4}-\frac{63817}{2660}a^{3}+\frac{11959}{1064}a^{2}-\frac{17397}{665}a+\frac{10715}{1064}$, $\frac{17}{1330}a^{11}+\frac{7}{190}a^{10}+\frac{13}{380}a^{9}-\frac{179}{1330}a^{8}-\frac{51}{133}a^{7}-\frac{9}{266}a^{6}-\frac{253}{266}a^{5}-\frac{1031}{266}a^{4}-\frac{16363}{1330}a^{3}-\frac{5328}{665}a^{2}-\frac{3169}{532}a+\frac{536}{133}$ | 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: | \( 68774.8668433 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 68774.8668433 \cdot 1}{2\cdot\sqrt{1740631100058763264}}\cr\approx \mathstrut & 0.649958831479 \end{aligned}\]
Galois group
$S_3\times D_6$ (as 12T37):
A solvable group of order 72 |
The 18 conjugacy class representatives for $S_3\times D_6$ |
Character table for $S_3\times D_6$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{2}, \sqrt{11})\), 6.2.41229056.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 36 siblings: | data not computed |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.26.45 | $x^{12} + 4 x^{11} + 2 x^{10} + 6 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 14$ | $12$ | $1$ | $26$ | $S_3 \times C_2^2$ | $[2, 3]_{3}^{2}$ |
\(11\) | 11.12.10.1 | $x^{12} + 42 x^{11} + 747 x^{10} + 7280 x^{9} + 41955 x^{8} + 143682 x^{7} + 279531 x^{6} + 287826 x^{5} + 175245 x^{4} + 124460 x^{3} + 344757 x^{2} + 893466 x + 996620$ | $6$ | $2$ | $10$ | $D_6$ | $[\ ]_{6}^{2}$ |