Normalized defining polynomial
\( x^{12} - 4x^{11} + 6x^{10} + 2x^{9} + 9x^{8} - 30x^{7} - 32x^{5} + 22x^{4} + 46x^{3} + 24x^{2} + 4x + 1 \)
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: | \(38806720086016\) \(\medspace = 2^{18}\cdot 23^{6}\) | 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: | \(13.56\) | 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: | $2^{7/4}23^{1/2}\approx 16.131190144457708$ | ||
Ramified primes: | \(2\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{746275793}a^{11}-\frac{1927181}{746275793}a^{10}-\frac{195723718}{746275793}a^{9}+\frac{342249133}{746275793}a^{8}+\frac{306533314}{746275793}a^{7}-\frac{243769531}{746275793}a^{6}-\frac{294731650}{746275793}a^{5}-\frac{150585591}{746275793}a^{4}+\frac{73606926}{746275793}a^{3}+\frac{20457170}{746275793}a^{2}-\frac{329916462}{746275793}a-\frac{154977604}{746275793}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{211159632}{746275793}a^{11}-\frac{879661871}{746275793}a^{10}+\frac{1420321783}{746275793}a^{9}+\frac{198810277}{746275793}a^{8}+\frac{1777019617}{746275793}a^{7}-\frac{6413623303}{746275793}a^{6}+\frac{1211125268}{746275793}a^{5}-\frac{6601254067}{746275793}a^{4}+\frac{5315597559}{746275793}a^{3}+\frac{8823309823}{746275793}a^{2}+\frac{2094145782}{746275793}a-\frac{148735221}{746275793}$, $\frac{141469928}{746275793}a^{11}-\frac{475580485}{746275793}a^{10}+\frac{453848565}{746275793}a^{9}+\frac{936797539}{746275793}a^{8}+\frac{1329045896}{746275793}a^{7}-\frac{3598418065}{746275793}a^{6}-\frac{3156025371}{746275793}a^{5}-\frac{3629780364}{746275793}a^{4}+\frac{527413691}{746275793}a^{3}+\frac{9378971830}{746275793}a^{2}+\frac{6845990825}{746275793}a+\frac{1862380028}{746275793}$, $\frac{104160803}{746275793}a^{11}-\frac{472582031}{746275793}a^{10}+\frac{838150449}{746275793}a^{9}-\frac{106060498}{746275793}a^{8}+\frac{812856603}{746275793}a^{7}-\frac{3709476419}{746275793}a^{6}+\frac{1514786817}{746275793}a^{5}-\frac{3402884523}{746275793}a^{4}+\frac{4098592504}{746275793}a^{3}+\frac{3684139919}{746275793}a^{2}+\frac{826272758}{746275793}a-\frac{2229274}{746275793}$, $\frac{160654467}{746275793}a^{11}-\frac{559298238}{746275793}a^{10}+\frac{640788824}{746275793}a^{9}+\frac{885545903}{746275793}a^{8}+\frac{1382063183}{746275793}a^{7}-\frac{3758691818}{746275793}a^{6}-\frac{1860953274}{746275793}a^{5}-\frac{3834393476}{746275793}a^{4}-\frac{272272276}{746275793}a^{3}+\frac{8046506725}{746275793}a^{2}+\frac{4227603410}{746275793}a+\frac{435212781}{746275793}$, $\frac{189771046}{746275793}a^{11}-\frac{1000256367}{746275793}a^{10}+\frac{2277276206}{746275793}a^{9}-\frac{1830869696}{746275793}a^{8}+\frac{2562420367}{746275793}a^{7}-\frac{8007027943}{746275793}a^{6}+\frac{8799851250}{746275793}a^{5}-\frac{11850387756}{746275793}a^{4}+\frac{14094527997}{746275793}a^{3}-\frac{2604219069}{746275793}a^{2}-\frac{106328429}{746275793}a-\frac{184170202}{746275793}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 45.0203706198 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 45.0203706198 \cdot 2}{2\cdot\sqrt{38806720086016}}\cr\approx \mathstrut & 0.444666904379 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T24):
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
\(\Q(\sqrt{23}) \), 3.1.23.1, 6.0.33856.1, 6.2.778688.2, 6.0.778688.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.778688.4, 6.0.33856.1 |
Degree 8 siblings: | 8.0.8667136.1, 8.4.4584914944.1 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.33856.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.51 | $x^{12} + 2 x^{11} + 16 x^{10} + 44 x^{9} + 18 x^{8} - 8 x^{7} + 24 x^{6} + 40 x^{5} + 20 x^{4} + 8 x^{3} + 8$ | $4$ | $3$ | $18$ | $A_4 \times C_2$ | $[2, 2, 2]^{3}$ |
\(23\) | 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |