Normalized defining polynomial
\( x^{12} - 4 x^{11} + 16 x^{10} - 36 x^{9} + 73 x^{8} - 104 x^{7} + 134 x^{6} - 132 x^{5} + 119 x^{4} + \cdots + 17 \)
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: | \(35664401793024\) \(\medspace = 2^{26}\cdot 3^{12}\) | 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: | \(13.47\) | 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: | $2^{13/6}3^{11/6}\approx 33.64758954662642$ | ||
Ramified primes: | \(2\), \(3\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13}a^{10}+\frac{5}{13}a^{9}+\frac{3}{13}a^{8}+\frac{3}{13}a^{6}+\frac{1}{13}a^{5}-\frac{5}{13}a^{4}-\frac{1}{13}a^{3}-\frac{3}{13}a^{2}-\frac{5}{13}a+\frac{5}{13}$, $\frac{1}{48477}a^{11}+\frac{443}{16159}a^{10}+\frac{11485}{48477}a^{9}+\frac{13111}{48477}a^{8}-\frac{19432}{48477}a^{7}-\frac{5414}{16159}a^{6}+\frac{22496}{48477}a^{5}+\frac{118}{429}a^{4}+\frac{9385}{48477}a^{3}+\frac{53}{143}a^{2}-\frac{118}{4407}a-\frac{7472}{48477}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{3200}{48477} a^{11} - \frac{4392}{16159} a^{10} + \frac{54911}{48477} a^{9} - \frac{122836}{48477} a^{8} + \frac{255976}{48477} a^{7} - \frac{115465}{16159} a^{6} + \frac{435148}{48477} a^{5} - \frac{3352}{429} a^{4} + \frac{315599}{48477} a^{3} - \frac{570}{143} a^{2} + \frac{10216}{4407} a - \frac{59716}{48477} \) (order $8$) | 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: | $\frac{4229}{48477}a^{11}-\frac{5969}{16159}a^{10}+\frac{66962}{48477}a^{9}-\frac{11767}{3729}a^{8}+\frac{281449}{48477}a^{7}-\frac{126532}{16159}a^{6}+\frac{396610}{48477}a^{5}-\frac{2677}{429}a^{4}+\frac{146849}{48477}a^{3}-\frac{98}{143}a^{2}-\frac{3065}{4407}a+\frac{2330}{3729}$, $\frac{109}{3729}a^{11}-\frac{190}{1243}a^{10}+\frac{2650}{3729}a^{9}-\frac{6566}{3729}a^{8}+\frac{14900}{3729}a^{7}-\frac{7159}{1243}a^{6}+\frac{28214}{3729}a^{5}-\frac{239}{33}a^{4}+\frac{23593}{3729}a^{3}-\frac{53}{11}a^{2}+\frac{698}{339}a-\frac{1526}{3729}$, $\frac{151}{1469}a^{11}-\frac{574}{1469}a^{10}+\frac{2284}{1469}a^{9}-\frac{4858}{1469}a^{8}+\frac{9644}{1469}a^{7}-\frac{12533}{1469}a^{6}+\frac{15258}{1469}a^{5}-\frac{109}{13}a^{4}+\frac{9833}{1469}a^{3}-\frac{41}{13}a^{2}+\frac{2317}{1469}a-\frac{1549}{1469}$, $\frac{8747}{48477}a^{11}-\frac{9454}{16159}a^{10}+\frac{115634}{48477}a^{9}-\frac{216031}{48477}a^{8}+\frac{424951}{48477}a^{7}-\frac{158305}{16159}a^{6}+\frac{567448}{48477}a^{5}-\frac{3493}{429}a^{4}+\frac{328541}{48477}a^{3}-\frac{565}{143}a^{2}+\frac{7567}{4407}a-\frac{103813}{48477}$, $\frac{3401}{48477}a^{11}-\frac{2359}{16159}a^{10}+\frac{40229}{48477}a^{9}-\frac{64264}{48477}a^{8}+\frac{179827}{48477}a^{7}-\frac{75035}{16159}a^{6}+\frac{381361}{48477}a^{5}-\frac{2833}{429}a^{4}+\frac{330026}{48477}a^{3}-\frac{477}{143}a^{2}+\frac{8194}{4407}a-\frac{6595}{48477}$ | 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: | \( 423.924544225 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 423.924544225 \cdot 1}{8\cdot\sqrt{35664401793024}}\cr\approx \mathstrut & 0.545959356392 \end{aligned}\]
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
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 6.0.1492992.5 |
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 | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.26.64 | $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$ | $12$ | $1$ | $26$ | $S_3 \times C_2^2$ | $[2, 3]_{3}^{2}$ |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.6.10.1 | $x^{6} + 18 x^{4} + 6 x^{3} + 162 x^{2} + 216 x + 90$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ |