Normalized defining polynomial
\( x^{12} - 6x^{10} + 8x^{8} - 38x^{6} + 114x^{4} - 76x^{2} + 38 \)
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: | \(479893735727955968\) \(\medspace = 2^{29}\cdot 19^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.75\) | 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: | $2^{10/3}19^{7/8}\approx 132.53870402717277$ | ||
Ramified primes: | \(2\), \(19\) | 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: | \(\Q(\sqrt{38}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{63101}a^{10}-\frac{1761}{63101}a^{8}-\frac{1386}{63101}a^{6}-\frac{28547}{63101}a^{4}-\frac{2095}{63101}a^{2}+\frac{16791}{63101}$, $\frac{1}{63101}a^{11}-\frac{1761}{63101}a^{9}-\frac{1386}{63101}a^{7}-\frac{28547}{63101}a^{5}-\frac{2095}{63101}a^{3}+\frac{16791}{63101}a$
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: | $\frac{2085}{63101}a^{10}-\frac{11827}{63101}a^{8}+\frac{12836}{63101}a^{6}-\frac{79353}{63101}a^{4}+\frac{238298}{63101}a^{2}-\frac{74921}{63101}$, $\frac{143}{63101}a^{10}+\frac{581}{63101}a^{8}-\frac{8895}{63101}a^{6}+\frac{19344}{63101}a^{4}-\frac{47181}{63101}a^{2}+\frac{3275}{63101}$, $\frac{540}{63101}a^{10}-\frac{4425}{63101}a^{8}+\frac{8772}{63101}a^{6}-\frac{18736}{63101}a^{4}+\frac{67619}{63101}a^{2}-\frac{82505}{63101}$, $\frac{6398}{63101}a^{11}-\frac{5393}{63101}a^{10}-\frac{34900}{63101}a^{9}+\frac{31923}{63101}a^{8}+\frac{29613}{63101}a^{7}-\frac{34321}{63101}a^{6}-\frac{218715}{63101}a^{5}+\frac{176834}{63101}a^{4}+\frac{604612}{63101}a^{3}-\frac{564653}{63101}a^{2}-\frac{32185}{63101}a+\frac{185375}{63101}$, $\frac{4530}{63101}a^{11}+\frac{1619}{63101}a^{10}-\frac{26604}{63101}a^{9}-\frac{11514}{63101}a^{8}+\frac{31520}{63101}a^{7}+\frac{27702}{63101}a^{6}-\frac{150163}{63101}a^{5}-\frac{90762}{63101}a^{4}+\frac{416507}{63101}a^{3}+\frac{204952}{63101}a^{2}-\frac{99677}{63101}a-\frac{75003}{63101}$, $\frac{2228}{63101}a^{11}+\frac{3450}{63101}a^{10}-\frac{11246}{63101}a^{9}-\frac{17754}{63101}a^{8}+\frac{3941}{63101}a^{7}+\frac{13976}{63101}a^{6}-\frac{60009}{63101}a^{5}-\frac{112691}{63101}a^{4}+\frac{191117}{63101}a^{3}+\frac{281269}{63101}a^{2}+\frac{54556}{63101}a-\frac{60869}{63101}$, $\frac{2228}{63101}a^{11}+\frac{2948}{63101}a^{10}-\frac{11246}{63101}a^{9}-\frac{17146}{63101}a^{8}+\frac{3941}{63101}a^{7}+\frac{15637}{63101}a^{6}-\frac{60009}{63101}a^{5}-\frac{106024}{63101}a^{4}+\frac{191117}{63101}a^{3}+\frac{323343}{63101}a^{2}+\frac{117657}{63101}a-\frac{34417}{63101}$ | 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: | \( 27252.2576419 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 27252.2576419 \cdot 1}{2\cdot\sqrt{479893735727955968}}\cr\approx \mathstrut & 0.490500445372 \end{aligned}\]
Galois group
$C_2\wr (C_2\times S_4)$ (as 12T250):
A solvable group of order 3072 |
The 65 conjugacy class representatives for $C_2\wr (C_2\times S_4)$ are not computed |
Character table for $C_2\wr (C_2\times S_4)$ is not computed |
Intermediate fields
3.1.76.1, 6.2.1755904.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 24 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 | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.29.110 | $x^{12} + 2 x^{10} + 2 x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{4} + 4 x^{2} + 14$ | $12$ | $1$ | $29$ | 12T193 | $[2, 8/3, 8/3, 3, 10/3, 10/3, 7/2]_{3}^{2}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.8.7.1 | $x^{8} + 19$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |